%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU813^1 : TPTP v6.1.0. Released v3.7.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n190.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:12 EDT 2014

% Result   : Timeout 300.10s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU813^1 : TPTP v6.1.0. Released v3.7.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n190.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:32:31 CDT 2014
% % CPUTime  : 300.10 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0xbce0e0>, <kernel.DependentProduct object at 0xbcec20>) of role type named in_type
% Using role type
% Declaring in:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x8b5e18>, <kernel.DependentProduct object at 0xbce710>) of role type named exu_type
% Using role type
% Declaring exu:((fofType->Prop)->Prop)
% FOF formula (((eq ((fofType->Prop)->Prop)) exu) (fun (Xphi:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))))) of role definition named exu
% A new definition: (((eq ((fofType->Prop)->Prop)) exu) (fun (Xphi:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))))
% Defined: exu:=(fun (Xphi:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))))
% FOF formula (<kernel.Constant object at 0xe27170>, <kernel.Sort object at 0xc80f38>) of role type named setextAx_type
% Using role type
% Declaring setextAx:Prop
% FOF formula (((eq Prop) setextAx) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), ((iff ((in Xx) A)) ((in Xx) B)))->(((eq fofType) A) B)))) of role definition named setextAx
% A new definition: (((eq Prop) setextAx) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), ((iff ((in Xx) A)) ((in Xx) B)))->(((eq fofType) A) B))))
% Defined: setextAx:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), ((iff ((in Xx) A)) ((in Xx) B)))->(((eq fofType) A) B)))
% FOF formula (<kernel.Constant object at 0xe27200>, <kernel.Single object at 0xbce710>) of role type named emptyset_type
% Using role type
% Declaring emptyset:fofType
% FOF formula (<kernel.Constant object at 0xbcec20>, <kernel.Sort object at 0xc80f38>) of role type named emptysetAx_type
% Using role type
% Declaring emptysetAx:Prop
% FOF formula (((eq Prop) emptysetAx) (forall (Xx:fofType), (((in Xx) emptyset)->False))) of role definition named emptysetAx
% A new definition: (((eq Prop) emptysetAx) (forall (Xx:fofType), (((in Xx) emptyset)->False)))
% Defined: emptysetAx:=(forall (Xx:fofType), (((in Xx) emptyset)->False))
% FOF formula (<kernel.Constant object at 0xbcee60>, <kernel.DependentProduct object at 0xbceb90>) of role type named setadjoin_type
% Using role type
% Declaring setadjoin:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0xbceb00>, <kernel.Sort object at 0xc80f38>) of role type named setadjoinAx_type
% Using role type
% Declaring setadjoinAx:Prop
% FOF formula (((eq Prop) setadjoinAx) (forall (Xx:fofType) (A:fofType) (Xy:fofType), ((iff ((in Xy) ((setadjoin Xx) A))) ((or (((eq fofType) Xy) Xx)) ((in Xy) A))))) of role definition named setadjoinAx
% A new definition: (((eq Prop) setadjoinAx) (forall (Xx:fofType) (A:fofType) (Xy:fofType), ((iff ((in Xy) ((setadjoin Xx) A))) ((or (((eq fofType) Xy) Xx)) ((in Xy) A)))))
% Defined: setadjoinAx:=(forall (Xx:fofType) (A:fofType) (Xy:fofType), ((iff ((in Xy) ((setadjoin Xx) A))) ((or (((eq fofType) Xy) Xx)) ((in Xy) A))))
% FOF formula (<kernel.Constant object at 0xbcec20>, <kernel.DependentProduct object at 0xbce950>) of role type named powerset_type
% Using role type
% Declaring powerset:(fofType->fofType)
% FOF formula (<kernel.Constant object at 0xbce560>, <kernel.Sort object at 0xc80f38>) of role type named powersetAx_type
% Using role type
% Declaring powersetAx:Prop
% FOF formula (((eq Prop) powersetAx) (forall (A:fofType) (B:fofType), ((iff ((in B) (powerset A))) (forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))))) of role definition named powersetAx
% A new definition: (((eq Prop) powersetAx) (forall (A:fofType) (B:fofType), ((iff ((in B) (powerset A))) (forall (Xx:fofType), (((in Xx) B)->((in Xx) A))))))
% Defined: powersetAx:=(forall (A:fofType) (B:fofType), ((iff ((in B) (powerset A))) (forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))))
% FOF formula (<kernel.Constant object at 0xbce128>, <kernel.DependentProduct object at 0xbcea28>) of role type named setunion_type
% Using role type
% Declaring setunion:(fofType->fofType)
% FOF formula (<kernel.Constant object at 0xbcec20>, <kernel.Sort object at 0xc80f38>) of role type named setunionAx_type
% Using role type
% Declaring setunionAx:Prop
% FOF formula (((eq Prop) setunionAx) (forall (A:fofType) (Xx:fofType), ((iff ((in Xx) (setunion A))) ((ex fofType) (fun (B:fofType)=> ((and ((in Xx) B)) ((in B) A))))))) of role definition named setunionAx
% A new definition: (((eq Prop) setunionAx) (forall (A:fofType) (Xx:fofType), ((iff ((in Xx) (setunion A))) ((ex fofType) (fun (B:fofType)=> ((and ((in Xx) B)) ((in B) A)))))))
% Defined: setunionAx:=(forall (A:fofType) (Xx:fofType), ((iff ((in Xx) (setunion A))) ((ex fofType) (fun (B:fofType)=> ((and ((in Xx) B)) ((in B) A))))))
% FOF formula (<kernel.Constant object at 0xbce560>, <kernel.Single object at 0xbcec20>) of role type named omega_type
% Using role type
% Declaring omega:fofType
% FOF formula (<kernel.Constant object at 0xbce248>, <kernel.Sort object at 0xc80f38>) of role type named omega0Ax_type
% Using role type
% Declaring omega0Ax:Prop
% FOF formula (((eq Prop) omega0Ax) ((in emptyset) omega)) of role definition named omega0Ax
% A new definition: (((eq Prop) omega0Ax) ((in emptyset) omega))
% Defined: omega0Ax:=((in emptyset) omega)
% FOF formula (<kernel.Constant object at 0xbce7e8>, <kernel.Sort object at 0xc80f38>) of role type named omegaSAx_type
% Using role type
% Declaring omegaSAx:Prop
% FOF formula (((eq Prop) omegaSAx) (forall (Xx:fofType), (((in Xx) omega)->((in ((setadjoin Xx) Xx)) omega)))) of role definition named omegaSAx
% A new definition: (((eq Prop) omegaSAx) (forall (Xx:fofType), (((in Xx) omega)->((in ((setadjoin Xx) Xx)) omega))))
% Defined: omegaSAx:=(forall (Xx:fofType), (((in Xx) omega)->((in ((setadjoin Xx) Xx)) omega)))
% FOF formula (<kernel.Constant object at 0xbceab8>, <kernel.Sort object at 0xc80f38>) of role type named omegaIndAx_type
% Using role type
% Declaring omegaIndAx:Prop
% FOF formula (((eq Prop) omegaIndAx) (forall (A:fofType), (((and ((in emptyset) A)) (forall (Xx:fofType), (((and ((in Xx) omega)) ((in Xx) A))->((in ((setadjoin Xx) Xx)) A))))->(forall (Xx:fofType), (((in Xx) omega)->((in Xx) A)))))) of role definition named omegaIndAx
% A new definition: (((eq Prop) omegaIndAx) (forall (A:fofType), (((and ((in emptyset) A)) (forall (Xx:fofType), (((and ((in Xx) omega)) ((in Xx) A))->((in ((setadjoin Xx) Xx)) A))))->(forall (Xx:fofType), (((in Xx) omega)->((in Xx) A))))))
% Defined: omegaIndAx:=(forall (A:fofType), (((and ((in emptyset) A)) (forall (Xx:fofType), (((and ((in Xx) omega)) ((in Xx) A))->((in ((setadjoin Xx) Xx)) A))))->(forall (Xx:fofType), (((in Xx) omega)->((in Xx) A)))))
% FOF formula (<kernel.Constant object at 0xbce2d8>, <kernel.Sort object at 0xc80f38>) of role type named replAx_type
% Using role type
% Declaring replAx:Prop
% FOF formula (((eq Prop) replAx) (forall (Xphi:(fofType->(fofType->Prop))) (A:fofType), ((forall (Xx:fofType), (((in Xx) A)->(exu (fun (Xy:fofType)=> ((Xphi Xx) Xy)))))->((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) ((Xphi Xy) Xx))))))))))) of role definition named replAx
% A new definition: (((eq Prop) replAx) (forall (Xphi:(fofType->(fofType->Prop))) (A:fofType), ((forall (Xx:fofType), (((in Xx) A)->(exu (fun (Xy:fofType)=> ((Xphi Xx) Xy)))))->((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) ((Xphi Xy) Xx)))))))))))
% Defined: replAx:=(forall (Xphi:(fofType->(fofType->Prop))) (A:fofType), ((forall (Xx:fofType), (((in Xx) A)->(exu (fun (Xy:fofType)=> ((Xphi Xx) Xy)))))->((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) ((Xphi Xy) Xx))))))))))
% FOF formula (<kernel.Constant object at 0xbce560>, <kernel.Sort object at 0xc80f38>) of role type named foundationAx_type
% Using role type
% Declaring foundationAx:Prop
% FOF formula (((eq Prop) foundationAx) (forall (A:fofType), (((ex fofType) (fun (Xx:fofType)=> ((in Xx) A)))->((ex fofType) (fun (B:fofType)=> ((and ((in B) A)) (((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) B)) ((in Xx) A))))->False))))))) of role definition named foundationAx
% A new definition: (((eq Prop) foundationAx) (forall (A:fofType), (((ex fofType) (fun (Xx:fofType)=> ((in Xx) A)))->((ex fofType) (fun (B:fofType)=> ((and ((in B) A)) (((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) B)) ((in Xx) A))))->False)))))))
% Defined: foundationAx:=(forall (A:fofType), (((ex fofType) (fun (Xx:fofType)=> ((in Xx) A)))->((ex fofType) (fun (B:fofType)=> ((and ((in B) A)) (((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) B)) ((in Xx) A))))->False))))))
% FOF formula (<kernel.Constant object at 0xbce488>, <kernel.Sort object at 0xc80f38>) of role type named wellorderingAx_type
% Using role type
% Declaring wellorderingAx:Prop
% FOF formula (((eq Prop) wellorderingAx) (forall (A:fofType), ((ex fofType) (fun (B:fofType)=> ((and ((and ((and (forall (C:fofType), (((in C) B)->(forall (Xx:fofType), (((in Xx) C)->((in Xx) A)))))) (forall (Xx:fofType) (Xy:fofType), (((and ((in Xx) A)) ((in Xy) A))->((forall (C:fofType), (((in C) B)->((iff ((in Xx) C)) ((in Xy) C))))->(((eq fofType) Xx) Xy)))))) (forall (C:fofType) (D:fofType), (((and ((in C) B)) ((in D) B))->((or (forall (Xx:fofType), (((in Xx) C)->((in Xx) D)))) (forall (Xx:fofType), (((in Xx) D)->((in Xx) C)))))))) (forall (C:fofType), (((and (forall (Xx:fofType), (((in Xx) C)->((in Xx) A)))) ((ex fofType) (fun (Xx:fofType)=> ((in Xx) C))))->((ex fofType) (fun (D:fofType)=> ((ex fofType) (fun (Xx:fofType)=> ((and ((and ((and ((in D) B)) ((in Xx) C))) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) D)) ((in Xy) C))))->False))) (forall (E:fofType), (((in E) B)->((or (forall (Xy:fofType), (((in Xy) E)->((in Xy) D)))) ((in Xx) E)))))))))))))))) of role definition named wellorderingAx
% A new definition: (((eq Prop) wellorderingAx) (forall (A:fofType), ((ex fofType) (fun (B:fofType)=> ((and ((and ((and (forall (C:fofType), (((in C) B)->(forall (Xx:fofType), (((in Xx) C)->((in Xx) A)))))) (forall (Xx:fofType) (Xy:fofType), (((and ((in Xx) A)) ((in Xy) A))->((forall (C:fofType), (((in C) B)->((iff ((in Xx) C)) ((in Xy) C))))->(((eq fofType) Xx) Xy)))))) (forall (C:fofType) (D:fofType), (((and ((in C) B)) ((in D) B))->((or (forall (Xx:fofType), (((in Xx) C)->((in Xx) D)))) (forall (Xx:fofType), (((in Xx) D)->((in Xx) C)))))))) (forall (C:fofType), (((and (forall (Xx:fofType), (((in Xx) C)->((in Xx) A)))) ((ex fofType) (fun (Xx:fofType)=> ((in Xx) C))))->((ex fofType) (fun (D:fofType)=> ((ex fofType) (fun (Xx:fofType)=> ((and ((and ((and ((in D) B)) ((in Xx) C))) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) D)) ((in Xy) C))))->False))) (forall (E:fofType), (((in E) B)->((or (forall (Xy:fofType), (((in Xy) E)->((in Xy) D)))) ((in Xx) E))))))))))))))))
% Defined: wellorderingAx:=(forall (A:fofType), ((ex fofType) (fun (B:fofType)=> ((and ((and ((and (forall (C:fofType), (((in C) B)->(forall (Xx:fofType), (((in Xx) C)->((in Xx) A)))))) (forall (Xx:fofType) (Xy:fofType), (((and ((in Xx) A)) ((in Xy) A))->((forall (C:fofType), (((in C) B)->((iff ((in Xx) C)) ((in Xy) C))))->(((eq fofType) Xx) Xy)))))) (forall (C:fofType) (D:fofType), (((and ((in C) B)) ((in D) B))->((or (forall (Xx:fofType), (((in Xx) C)->((in Xx) D)))) (forall (Xx:fofType), (((in Xx) D)->((in Xx) C)))))))) (forall (C:fofType), (((and (forall (Xx:fofType), (((in Xx) C)->((in Xx) A)))) ((ex fofType) (fun (Xx:fofType)=> ((in Xx) C))))->((ex fofType) (fun (D:fofType)=> ((ex fofType) (fun (Xx:fofType)=> ((and ((and ((and ((in D) B)) ((in Xx) C))) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) D)) ((in Xy) C))))->False))) (forall (E:fofType), (((in E) B)->((or (forall (Xy:fofType), (((in Xy) E)->((in Xy) D)))) ((in Xx) E)))))))))))))))
% FOF formula (<kernel.Constant object at 0xbce488>, <kernel.DependentProduct object at 0xe1ea28>) of role type named descr_type
% Using role type
% Declaring descr:((fofType->Prop)->fofType)
% FOF formula (<kernel.Constant object at 0xbce560>, <kernel.Sort object at 0xc80f38>) of role type named descrp_type
% Using role type
% Declaring descrp:Prop
% FOF formula (((eq Prop) descrp) (forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->(Xphi (descr (fun (Xx:fofType)=> (Xphi Xx))))))) of role definition named descrp
% A new definition: (((eq Prop) descrp) (forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->(Xphi (descr (fun (Xx:fofType)=> (Xphi Xx)))))))
% Defined: descrp:=(forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->(Xphi (descr (fun (Xx:fofType)=> (Xphi Xx))))))
% FOF formula (<kernel.Constant object at 0xbce488>, <kernel.DependentProduct object at 0xe1ee18>) of role type named dsetconstr_type
% Using role type
% Declaring dsetconstr:(fofType->((fofType->Prop)->fofType))
% FOF formula (<kernel.Constant object at 0xe1ed40>, <kernel.Sort object at 0xc80f38>) of role type named dsetconstrI_type
% Using role type
% Declaring dsetconstrI:Prop
% FOF formula (((eq Prop) dsetconstrI) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy)))))))) of role definition named dsetconstrI
% A new definition: (((eq Prop) dsetconstrI) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))))))
% Defined: dsetconstrI:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy)))))))
% FOF formula (<kernel.Constant object at 0xe1ea28>, <kernel.Sort object at 0xc80f38>) of role type named dsetconstrEL_type
% Using role type
% Declaring dsetconstrEL:Prop
% FOF formula (((eq Prop) dsetconstrEL) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))->((in Xx) A)))) of role definition named dsetconstrEL
% A new definition: (((eq Prop) dsetconstrEL) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))->((in Xx) A))))
% Defined: dsetconstrEL:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))->((in Xx) A)))
% FOF formula (<kernel.Constant object at 0xe1ea70>, <kernel.Sort object at 0xc80f38>) of role type named dsetconstrER_type
% Using role type
% Declaring dsetconstrER:Prop
% FOF formula (((eq Prop) dsetconstrER) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))->(Xphi Xx)))) of role definition named dsetconstrER
% A new definition: (((eq Prop) dsetconstrER) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))->(Xphi Xx))))
% Defined: dsetconstrER:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))->(Xphi Xx)))
% FOF formula (<kernel.Constant object at 0xe1ecf8>, <kernel.Sort object at 0xc80f38>) of role type named exuE1_type
% Using role type
% Declaring exuE1:Prop
% FOF formula (((eq Prop) exuE1) (forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))))) of role definition named exuE1
% A new definition: (((eq Prop) exuE1) (forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))))))
% Defined: exuE1:=(forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))))
% FOF formula (<kernel.Constant object at 0xe1e128>, <kernel.DependentProduct object at 0xbcf9e0>) of role type named prop2set_type
% Using role type
% Declaring prop2set:(Prop->fofType)
% FOF formula (<kernel.Constant object at 0xe1e050>, <kernel.Sort object at 0xc80f38>) of role type named prop2setE_type
% Using role type
% Declaring prop2setE:Prop
% FOF formula (((eq Prop) prop2setE) (forall (Xphi:Prop) (Xx:fofType), (((in Xx) (prop2set Xphi))->Xphi))) of role definition named prop2setE
% A new definition: (((eq Prop) prop2setE) (forall (Xphi:Prop) (Xx:fofType), (((in Xx) (prop2set Xphi))->Xphi)))
% Defined: prop2setE:=(forall (Xphi:Prop) (Xx:fofType), (((in Xx) (prop2set Xphi))->Xphi))
% FOF formula (<kernel.Constant object at 0xe1e128>, <kernel.Sort object at 0xc80f38>) of role type named emptysetE_type
% Using role type
% Declaring emptysetE:Prop
% FOF formula (((eq Prop) emptysetE) (forall (Xx:fofType), (((in Xx) emptyset)->(forall (Xphi:Prop), Xphi)))) of role definition named emptysetE
% A new definition: (((eq Prop) emptysetE) (forall (Xx:fofType), (((in Xx) emptyset)->(forall (Xphi:Prop), Xphi))))
% Defined: emptysetE:=(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Xphi:Prop), Xphi)))
% FOF formula (<kernel.Constant object at 0xe1e128>, <kernel.Sort object at 0xc80f38>) of role type named emptysetimpfalse_type
% Using role type
% Declaring emptysetimpfalse:Prop
% FOF formula (((eq Prop) emptysetimpfalse) (forall (Xx:fofType), (((in Xx) emptyset)->False))) of role definition named emptysetimpfalse
% A new definition: (((eq Prop) emptysetimpfalse) (forall (Xx:fofType), (((in Xx) emptyset)->False)))
% Defined: emptysetimpfalse:=(forall (Xx:fofType), (((in Xx) emptyset)->False))
% FOF formula (<kernel.Constant object at 0xe1ebd8>, <kernel.Sort object at 0xc80f38>) of role type named notinemptyset_type
% Using role type
% Declaring notinemptyset:Prop
% FOF formula (((eq Prop) notinemptyset) (forall (Xx:fofType), (((in Xx) emptyset)->False))) of role definition named notinemptyset
% A new definition: (((eq Prop) notinemptyset) (forall (Xx:fofType), (((in Xx) emptyset)->False)))
% Defined: notinemptyset:=(forall (Xx:fofType), (((in Xx) emptyset)->False))
% FOF formula (<kernel.Constant object at 0xbcf0e0>, <kernel.Sort object at 0xc80f38>) of role type named exuE3e_type
% Using role type
% Declaring exuE3e:Prop
% FOF formula (((eq Prop) exuE3e) (forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> (Xphi Xx)))))) of role definition named exuE3e
% A new definition: (((eq Prop) exuE3e) (forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> (Xphi Xx))))))
% Defined: exuE3e:=(forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> (Xphi Xx)))))
% FOF formula (<kernel.Constant object at 0xbcf878>, <kernel.Sort object at 0xc80f38>) of role type named setext_type
% Using role type
% Declaring setext:Prop
% FOF formula (((eq Prop) setext) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->(((eq fofType) A) B))))) of role definition named setext
% A new definition: (((eq Prop) setext) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->(((eq fofType) A) B)))))
% Defined: setext:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->(((eq fofType) A) B))))
% FOF formula (<kernel.Constant object at 0xbcf9e0>, <kernel.Sort object at 0xc80f38>) of role type named emptyI_type
% Using role type
% Declaring emptyI:Prop
% FOF formula (((eq Prop) emptyI) (forall (A:fofType), ((forall (Xx:fofType), (((in Xx) A)->False))->(((eq fofType) A) emptyset)))) of role definition named emptyI
% A new definition: (((eq Prop) emptyI) (forall (A:fofType), ((forall (Xx:fofType), (((in Xx) A)->False))->(((eq fofType) A) emptyset))))
% Defined: emptyI:=(forall (A:fofType), ((forall (Xx:fofType), (((in Xx) A)->False))->(((eq fofType) A) emptyset)))
% FOF formula (<kernel.Constant object at 0xbcfa28>, <kernel.Sort object at 0xc80f38>) of role type named noeltsimpempty_type
% Using role type
% Declaring noeltsimpempty:Prop
% FOF formula (((eq Prop) noeltsimpempty) (forall (A:fofType), ((forall (Xx:fofType), (((in Xx) A)->False))->(((eq fofType) A) emptyset)))) of role definition named noeltsimpempty
% A new definition: (((eq Prop) noeltsimpempty) (forall (A:fofType), ((forall (Xx:fofType), (((in Xx) A)->False))->(((eq fofType) A) emptyset))))
% Defined: noeltsimpempty:=(forall (A:fofType), ((forall (Xx:fofType), (((in Xx) A)->False))->(((eq fofType) A) emptyset)))
% FOF formula (<kernel.Constant object at 0xbcf2d8>, <kernel.Sort object at 0xc80f38>) of role type named setbeta_type
% Using role type
% Declaring setbeta:Prop
% FOF formula (((eq Prop) setbeta) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((iff ((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))) (Xphi Xx))))) of role definition named setbeta
% A new definition: (((eq Prop) setbeta) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((iff ((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))) (Xphi Xx)))))
% Defined: setbeta:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((iff ((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))) (Xphi Xx))))
% FOF formula (<kernel.Constant object at 0xbcfc20>, <kernel.DependentProduct object at 0xbcfdd0>) of role type named nonempty_type
% Using role type
% Declaring nonempty:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) nonempty) (fun (Xx:fofType)=> (not (((eq fofType) Xx) emptyset)))) of role definition named nonempty
% A new definition: (((eq (fofType->Prop)) nonempty) (fun (Xx:fofType)=> (not (((eq fofType) Xx) emptyset))))
% Defined: nonempty:=(fun (Xx:fofType)=> (not (((eq fofType) Xx) emptyset)))
% FOF formula (<kernel.Constant object at 0xbcfdd0>, <kernel.Sort object at 0xc80f38>) of role type named nonemptyE1_type
% Using role type
% Declaring nonemptyE1:Prop
% FOF formula (((eq Prop) nonemptyE1) (forall (A:fofType), ((nonempty A)->((ex fofType) (fun (Xx:fofType)=> ((in Xx) A)))))) of role definition named nonemptyE1
% A new definition: (((eq Prop) nonemptyE1) (forall (A:fofType), ((nonempty A)->((ex fofType) (fun (Xx:fofType)=> ((in Xx) A))))))
% Defined: nonemptyE1:=(forall (A:fofType), ((nonempty A)->((ex fofType) (fun (Xx:fofType)=> ((in Xx) A)))))
% FOF formula (<kernel.Constant object at 0xbcf710>, <kernel.Sort object at 0xc80f38>) of role type named nonemptyI_type
% Using role type
% Declaring nonemptyI:Prop
% FOF formula (((eq Prop) nonemptyI) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->(nonempty ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy)))))))) of role definition named nonemptyI
% A new definition: (((eq Prop) nonemptyI) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->(nonempty ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))))))
% Defined: nonemptyI:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->(nonempty ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy)))))))
% FOF formula (<kernel.Constant object at 0xbb0908>, <kernel.Sort object at 0xc80f38>) of role type named nonemptyI1_type
% Using role type
% Declaring nonemptyI1:Prop
% FOF formula (((eq Prop) nonemptyI1) (forall (A:fofType), (((ex fofType) (fun (Xx:fofType)=> ((in Xx) A)))->(nonempty A)))) of role definition named nonemptyI1
% A new definition: (((eq Prop) nonemptyI1) (forall (A:fofType), (((ex fofType) (fun (Xx:fofType)=> ((in Xx) A)))->(nonempty A))))
% Defined: nonemptyI1:=(forall (A:fofType), (((ex fofType) (fun (Xx:fofType)=> ((in Xx) A)))->(nonempty A)))
% FOF formula (<kernel.Constant object at 0xbb0908>, <kernel.Sort object at 0xc80f38>) of role type named setadjoinIL_type
% Using role type
% Declaring setadjoinIL:Prop
% FOF formula (((eq Prop) setadjoinIL) (forall (Xx:fofType) (Xy:fofType), ((in Xx) ((setadjoin Xx) Xy)))) of role definition named setadjoinIL
% A new definition: (((eq Prop) setadjoinIL) (forall (Xx:fofType) (Xy:fofType), ((in Xx) ((setadjoin Xx) Xy))))
% Defined: setadjoinIL:=(forall (Xx:fofType) (Xy:fofType), ((in Xx) ((setadjoin Xx) Xy)))
% FOF formula (<kernel.Constant object at 0xbb0908>, <kernel.Sort object at 0xc80f38>) of role type named emptyinunitempty_type
% Using role type
% Declaring emptyinunitempty:Prop
% FOF formula (((eq Prop) emptyinunitempty) ((in emptyset) ((setadjoin emptyset) emptyset))) of role definition named emptyinunitempty
% A new definition: (((eq Prop) emptyinunitempty) ((in emptyset) ((setadjoin emptyset) emptyset)))
% Defined: emptyinunitempty:=((in emptyset) ((setadjoin emptyset) emptyset))
% FOF formula (<kernel.Constant object at 0xbcfb48>, <kernel.Sort object at 0xc80f38>) of role type named setadjoinIR_type
% Using role type
% Declaring setadjoinIR:Prop
% FOF formula (((eq Prop) setadjoinIR) (forall (Xx:fofType) (A:fofType) (Xy:fofType), (((in Xy) A)->((in Xy) ((setadjoin Xx) A))))) of role definition named setadjoinIR
% A new definition: (((eq Prop) setadjoinIR) (forall (Xx:fofType) (A:fofType) (Xy:fofType), (((in Xy) A)->((in Xy) ((setadjoin Xx) A)))))
% Defined: setadjoinIR:=(forall (Xx:fofType) (A:fofType) (Xy:fofType), (((in Xy) A)->((in Xy) ((setadjoin Xx) A))))
% FOF formula (<kernel.Constant object at 0xbcfcf8>, <kernel.Sort object at 0xc80f38>) of role type named setadjoinE_type
% Using role type
% Declaring setadjoinE:Prop
% FOF formula (((eq Prop) setadjoinE) (forall (Xx:fofType) (A:fofType) (Xy:fofType), (((in Xy) ((setadjoin Xx) A))->(forall (Xphi:Prop), (((((eq fofType) Xy) Xx)->Xphi)->((((in Xy) A)->Xphi)->Xphi)))))) of role definition named setadjoinE
% A new definition: (((eq Prop) setadjoinE) (forall (Xx:fofType) (A:fofType) (Xy:fofType), (((in Xy) ((setadjoin Xx) A))->(forall (Xphi:Prop), (((((eq fofType) Xy) Xx)->Xphi)->((((in Xy) A)->Xphi)->Xphi))))))
% Defined: setadjoinE:=(forall (Xx:fofType) (A:fofType) (Xy:fofType), (((in Xy) ((setadjoin Xx) A))->(forall (Xphi:Prop), (((((eq fofType) Xy) Xx)->Xphi)->((((in Xy) A)->Xphi)->Xphi)))))
% FOF formula (<kernel.Constant object at 0xbcf998>, <kernel.Sort object at 0xc80f38>) of role type named setadjoinOr_type
% Using role type
% Declaring setadjoinOr:Prop
% FOF formula (((eq Prop) setadjoinOr) (forall (Xx:fofType) (A:fofType) (Xy:fofType), (((in Xy) ((setadjoin Xx) A))->((or (((eq fofType) Xy) Xx)) ((in Xy) A))))) of role definition named setadjoinOr
% A new definition: (((eq Prop) setadjoinOr) (forall (Xx:fofType) (A:fofType) (Xy:fofType), (((in Xy) ((setadjoin Xx) A))->((or (((eq fofType) Xy) Xx)) ((in Xy) A)))))
% Defined: setadjoinOr:=(forall (Xx:fofType) (A:fofType) (Xy:fofType), (((in Xy) ((setadjoin Xx) A))->((or (((eq fofType) Xy) Xx)) ((in Xy) A))))
% FOF formula (<kernel.Constant object at 0xbcf4d0>, <kernel.Sort object at 0xc80f38>) of role type named setoftrueEq_type
% Using role type
% Declaring setoftrueEq:Prop
% FOF formula (((eq Prop) setoftrueEq) (forall (A:fofType), (((eq fofType) ((dsetconstr A) (fun (Xx:fofType)=> True))) A))) of role definition named setoftrueEq
% A new definition: (((eq Prop) setoftrueEq) (forall (A:fofType), (((eq fofType) ((dsetconstr A) (fun (Xx:fofType)=> True))) A)))
% Defined: setoftrueEq:=(forall (A:fofType), (((eq fofType) ((dsetconstr A) (fun (Xx:fofType)=> True))) A))
% FOF formula (<kernel.Constant object at 0xbcfb00>, <kernel.Sort object at 0xc80f38>) of role type named powersetI_type
% Using role type
% Declaring powersetI:Prop
% FOF formula (((eq Prop) powersetI) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->((in B) (powerset A))))) of role definition named powersetI
% A new definition: (((eq Prop) powersetI) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->((in B) (powerset A)))))
% Defined: powersetI:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->((in B) (powerset A))))
% FOF formula (<kernel.Constant object at 0xbcf710>, <kernel.Sort object at 0xc80f38>) of role type named emptyinPowerset_type
% Using role type
% Declaring emptyinPowerset:Prop
% FOF formula (((eq Prop) emptyinPowerset) (forall (A:fofType), ((in emptyset) (powerset A)))) of role definition named emptyinPowerset
% A new definition: (((eq Prop) emptyinPowerset) (forall (A:fofType), ((in emptyset) (powerset A))))
% Defined: emptyinPowerset:=(forall (A:fofType), ((in emptyset) (powerset A)))
% FOF formula (<kernel.Constant object at 0xbcf710>, <kernel.Sort object at 0xc80f38>) of role type named emptyInPowerset_type
% Using role type
% Declaring emptyInPowerset:Prop
% FOF formula (((eq Prop) emptyInPowerset) (forall (A:fofType), ((in emptyset) (powerset A)))) of role definition named emptyInPowerset
% A new definition: (((eq Prop) emptyInPowerset) (forall (A:fofType), ((in emptyset) (powerset A))))
% Defined: emptyInPowerset:=(forall (A:fofType), ((in emptyset) (powerset A)))
% FOF formula (<kernel.Constant object at 0xbcf710>, <kernel.Sort object at 0xc80f38>) of role type named powersetE_type
% Using role type
% Declaring powersetE:Prop
% FOF formula (((eq Prop) powersetE) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in B) (powerset A))->(((in Xx) B)->((in Xx) A))))) of role definition named powersetE
% A new definition: (((eq Prop) powersetE) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in B) (powerset A))->(((in Xx) B)->((in Xx) A)))))
% Defined: powersetE:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in B) (powerset A))->(((in Xx) B)->((in Xx) A))))
% FOF formula (<kernel.Constant object at 0x9d29e0>, <kernel.Sort object at 0xc80f38>) of role type named setunionI_type
% Using role type
% Declaring setunionI:Prop
% FOF formula (((eq Prop) setunionI) (forall (A:fofType) (Xx:fofType) (B:fofType), (((in Xx) B)->(((in B) A)->((in Xx) (setunion A)))))) of role definition named setunionI
% A new definition: (((eq Prop) setunionI) (forall (A:fofType) (Xx:fofType) (B:fofType), (((in Xx) B)->(((in B) A)->((in Xx) (setunion A))))))
% Defined: setunionI:=(forall (A:fofType) (Xx:fofType) (B:fofType), (((in Xx) B)->(((in B) A)->((in Xx) (setunion A)))))
% FOF formula (<kernel.Constant object at 0x9d2560>, <kernel.Sort object at 0xc80f38>) of role type named setunionE_type
% Using role type
% Declaring setunionE:Prop
% FOF formula (((eq Prop) setunionE) (forall (A:fofType) (Xx:fofType), (((in Xx) (setunion A))->(forall (Xphi:Prop), ((forall (B:fofType), (((in Xx) B)->(((in B) A)->Xphi)))->Xphi))))) of role definition named setunionE
% A new definition: (((eq Prop) setunionE) (forall (A:fofType) (Xx:fofType), (((in Xx) (setunion A))->(forall (Xphi:Prop), ((forall (B:fofType), (((in Xx) B)->(((in B) A)->Xphi)))->Xphi)))))
% Defined: setunionE:=(forall (A:fofType) (Xx:fofType), (((in Xx) (setunion A))->(forall (Xphi:Prop), ((forall (B:fofType), (((in Xx) B)->(((in B) A)->Xphi)))->Xphi))))
% FOF formula (<kernel.Constant object at 0x9d26c8>, <kernel.Sort object at 0xc80f38>) of role type named subPowSU_type
% Using role type
% Declaring subPowSU:Prop
% FOF formula (((eq Prop) subPowSU) (forall (A:fofType) (Xx:fofType), (((in Xx) A)->((in Xx) (powerset (setunion A)))))) of role definition named subPowSU
% A new definition: (((eq Prop) subPowSU) (forall (A:fofType) (Xx:fofType), (((in Xx) A)->((in Xx) (powerset (setunion A))))))
% Defined: subPowSU:=(forall (A:fofType) (Xx:fofType), (((in Xx) A)->((in Xx) (powerset (setunion A)))))
% FOF formula (<kernel.Constant object at 0x9d2518>, <kernel.Sort object at 0xc80f38>) of role type named exuE2_type
% Using role type
% Declaring exuE2:Prop
% FOF formula (((eq Prop) exuE2) (forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))))) of role definition named exuE2
% A new definition: (((eq Prop) exuE2) (forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))))))
% Defined: exuE2:=(forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))))
% FOF formula (<kernel.Constant object at 0x9d23f8>, <kernel.Sort object at 0xc80f38>) of role type named nonemptyImpWitness_type
% Using role type
% Declaring nonemptyImpWitness:Prop
% FOF formula (((eq Prop) nonemptyImpWitness) (forall (A:fofType), ((nonempty A)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))))) of role definition named nonemptyImpWitness
% A new definition: (((eq Prop) nonemptyImpWitness) (forall (A:fofType), ((nonempty A)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))))))
% Defined: nonemptyImpWitness:=(forall (A:fofType), ((nonempty A)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) True)))))
% FOF formula (<kernel.Constant object at 0x9d2908>, <kernel.Sort object at 0xc80f38>) of role type named uniqinunit_type
% Using role type
% Declaring uniqinunit:Prop
% FOF formula (((eq Prop) uniqinunit) (forall (Xx:fofType) (Xy:fofType), (((in Xx) ((setadjoin Xy) emptyset))->(((eq fofType) Xx) Xy)))) of role definition named uniqinunit
% A new definition: (((eq Prop) uniqinunit) (forall (Xx:fofType) (Xy:fofType), (((in Xx) ((setadjoin Xy) emptyset))->(((eq fofType) Xx) Xy))))
% Defined: uniqinunit:=(forall (Xx:fofType) (Xy:fofType), (((in Xx) ((setadjoin Xy) emptyset))->(((eq fofType) Xx) Xy)))
% FOF formula (<kernel.Constant object at 0x9d2f38>, <kernel.Sort object at 0xc80f38>) of role type named notinsingleton_type
% Using role type
% Declaring notinsingleton:Prop
% FOF formula (((eq Prop) notinsingleton) (forall (Xx:fofType) (Xy:fofType), ((not (((eq fofType) Xx) Xy))->(((in Xy) ((setadjoin Xx) emptyset))->False)))) of role definition named notinsingleton
% A new definition: (((eq Prop) notinsingleton) (forall (Xx:fofType) (Xy:fofType), ((not (((eq fofType) Xx) Xy))->(((in Xy) ((setadjoin Xx) emptyset))->False))))
% Defined: notinsingleton:=(forall (Xx:fofType) (Xy:fofType), ((not (((eq fofType) Xx) Xy))->(((in Xy) ((setadjoin Xx) emptyset))->False)))
% FOF formula (<kernel.Constant object at 0x9d24d0>, <kernel.Sort object at 0xc80f38>) of role type named eqinunit_type
% Using role type
% Declaring eqinunit:Prop
% FOF formula (((eq Prop) eqinunit) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((in Xx) ((setadjoin Xy) emptyset))))) of role definition named eqinunit
% A new definition: (((eq Prop) eqinunit) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((in Xx) ((setadjoin Xy) emptyset)))))
% Defined: eqinunit:=(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((in Xx) ((setadjoin Xy) emptyset))))
% FOF formula (<kernel.Constant object at 0x9d2ef0>, <kernel.Sort object at 0xc80f38>) of role type named singletonsswitch_type
% Using role type
% Declaring singletonsswitch:Prop
% FOF formula (((eq Prop) singletonsswitch) (forall (Xx:fofType) (Xy:fofType), (((in Xx) ((setadjoin Xy) emptyset))->((in Xy) ((setadjoin Xx) emptyset))))) of role definition named singletonsswitch
% A new definition: (((eq Prop) singletonsswitch) (forall (Xx:fofType) (Xy:fofType), (((in Xx) ((setadjoin Xy) emptyset))->((in Xy) ((setadjoin Xx) emptyset)))))
% Defined: singletonsswitch:=(forall (Xx:fofType) (Xy:fofType), (((in Xx) ((setadjoin Xy) emptyset))->((in Xy) ((setadjoin Xx) emptyset))))
% FOF formula (<kernel.Constant object at 0x9d25f0>, <kernel.Sort object at 0xc80f38>) of role type named upairsetE_type
% Using role type
% Declaring upairsetE:Prop
% FOF formula (((eq Prop) upairsetE) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((in Xz) ((setadjoin Xx) ((setadjoin Xy) emptyset)))->((or (((eq fofType) Xz) Xx)) (((eq fofType) Xz) Xy))))) of role definition named upairsetE
% A new definition: (((eq Prop) upairsetE) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((in Xz) ((setadjoin Xx) ((setadjoin Xy) emptyset)))->((or (((eq fofType) Xz) Xx)) (((eq fofType) Xz) Xy)))))
% Defined: upairsetE:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((in Xz) ((setadjoin Xx) ((setadjoin Xy) emptyset)))->((or (((eq fofType) Xz) Xx)) (((eq fofType) Xz) Xy))))
% FOF formula (<kernel.Constant object at 0x9d2fc8>, <kernel.Sort object at 0xc80f38>) of role type named upairsetIL_type
% Using role type
% Declaring upairsetIL:Prop
% FOF formula (((eq Prop) upairsetIL) (forall (Xx:fofType) (Xy:fofType), ((in Xx) ((setadjoin Xx) ((setadjoin Xy) emptyset))))) of role definition named upairsetIL
% A new definition: (((eq Prop) upairsetIL) (forall (Xx:fofType) (Xy:fofType), ((in Xx) ((setadjoin Xx) ((setadjoin Xy) emptyset)))))
% Defined: upairsetIL:=(forall (Xx:fofType) (Xy:fofType), ((in Xx) ((setadjoin Xx) ((setadjoin Xy) emptyset))))
% FOF formula (<kernel.Constant object at 0x9d2bd8>, <kernel.Sort object at 0xc80f38>) of role type named upairsetIR_type
% Using role type
% Declaring upairsetIR:Prop
% FOF formula (((eq Prop) upairsetIR) (forall (Xx:fofType) (Xy:fofType), ((in Xy) ((setadjoin Xx) ((setadjoin Xy) emptyset))))) of role definition named upairsetIR
% A new definition: (((eq Prop) upairsetIR) (forall (Xx:fofType) (Xy:fofType), ((in Xy) ((setadjoin Xx) ((setadjoin Xy) emptyset)))))
% Defined: upairsetIR:=(forall (Xx:fofType) (Xy:fofType), ((in Xy) ((setadjoin Xx) ((setadjoin Xy) emptyset))))
% FOF formula (<kernel.Constant object at 0x9d2878>, <kernel.Sort object at 0xc80f38>) of role type named emptyE1_type
% Using role type
% Declaring emptyE1:Prop
% FOF formula (((eq Prop) emptyE1) (forall (A:fofType) (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->((((eq fofType) ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) emptyset)->False)))) of role definition named emptyE1
% A new definition: (((eq Prop) emptyE1) (forall (A:fofType) (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->((((eq fofType) ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) emptyset)->False))))
% Defined: emptyE1:=(forall (A:fofType) (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->((((eq fofType) ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) emptyset)->False)))
% FOF formula (<kernel.Constant object at 0x9d2b48>, <kernel.Sort object at 0xc80f38>) of role type named vacuousDall_type
% Using role type
% Declaring vacuousDall:Prop
% FOF formula (((eq Prop) vacuousDall) (forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx)))) of role definition named vacuousDall
% A new definition: (((eq Prop) vacuousDall) (forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx))))
% Defined: vacuousDall:=(forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx)))
% FOF formula (<kernel.Constant object at 0x9d2b90>, <kernel.Sort object at 0xc80f38>) of role type named quantDeMorgan1_type
% Using role type
% Declaring quantDeMorgan1:Prop
% FOF formula (((eq Prop) quantDeMorgan1) (forall (A:fofType) (Xphi:(fofType->Prop)), (((forall (Xx:fofType), (((in Xx) A)->(Xphi Xx)))->False)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((Xphi Xx)->False))))))) of role definition named quantDeMorgan1
% A new definition: (((eq Prop) quantDeMorgan1) (forall (A:fofType) (Xphi:(fofType->Prop)), (((forall (Xx:fofType), (((in Xx) A)->(Xphi Xx)))->False)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((Xphi Xx)->False)))))))
% Defined: quantDeMorgan1:=(forall (A:fofType) (Xphi:(fofType->Prop)), (((forall (Xx:fofType), (((in Xx) A)->(Xphi Xx)))->False)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((Xphi Xx)->False))))))
% FOF formula (<kernel.Constant object at 0x9d2b00>, <kernel.Sort object at 0xc80f38>) of role type named quantDeMorgan2_type
% Using role type
% Declaring quantDeMorgan2:Prop
% FOF formula (((eq Prop) quantDeMorgan2) (forall (A:fofType) (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->((Xphi Xx)->False)))->(((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->False)))) of role definition named quantDeMorgan2
% A new definition: (((eq Prop) quantDeMorgan2) (forall (A:fofType) (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->((Xphi Xx)->False)))->(((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->False))))
% Defined: quantDeMorgan2:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->((Xphi Xx)->False)))->(((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->False)))
% FOF formula (<kernel.Constant object at 0x9d2638>, <kernel.Sort object at 0xc80f38>) of role type named quantDeMorgan3_type
% Using role type
% Declaring quantDeMorgan3:Prop
% FOF formula (((eq Prop) quantDeMorgan3) (forall (A:fofType) (Xphi:(fofType->Prop)), ((((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->False)->(forall (Xx:fofType), (((in Xx) A)->((Xphi Xx)->False)))))) of role definition named quantDeMorgan3
% A new definition: (((eq Prop) quantDeMorgan3) (forall (A:fofType) (Xphi:(fofType->Prop)), ((((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->False)->(forall (Xx:fofType), (((in Xx) A)->((Xphi Xx)->False))))))
% Defined: quantDeMorgan3:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->False)->(forall (Xx:fofType), (((in Xx) A)->((Xphi Xx)->False)))))
% FOF formula (<kernel.Constant object at 0x9d2f80>, <kernel.Sort object at 0xc80f38>) of role type named quantDeMorgan4_type
% Using role type
% Declaring quantDeMorgan4:Prop
% FOF formula (((eq Prop) quantDeMorgan4) (forall (A:fofType) (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((Xphi Xx)->False))))->((forall (Xx:fofType), (((in Xx) A)->(Xphi Xx)))->False)))) of role definition named quantDeMorgan4
% A new definition: (((eq Prop) quantDeMorgan4) (forall (A:fofType) (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((Xphi Xx)->False))))->((forall (Xx:fofType), (((in Xx) A)->(Xphi Xx)))->False))))
% Defined: quantDeMorgan4:=(forall (A:fofType) (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((Xphi Xx)->False))))->((forall (Xx:fofType), (((in Xx) A)->(Xphi Xx)))->False)))
% FOF formula (<kernel.Constant object at 0x9d2dd0>, <kernel.Sort object at 0xc80f38>) of role type named prop2setI_type
% Using role type
% Declaring prop2setI:Prop
% FOF formula (((eq Prop) prop2setI) (forall (Xphi:Prop), (Xphi->((in emptyset) (prop2set Xphi))))) of role definition named prop2setI
% A new definition: (((eq Prop) prop2setI) (forall (Xphi:Prop), (Xphi->((in emptyset) (prop2set Xphi)))))
% Defined: prop2setI:=(forall (Xphi:Prop), (Xphi->((in emptyset) (prop2set Xphi))))
% FOF formula (<kernel.Constant object at 0x9d2dd0>, <kernel.DependentProduct object at 0xbbe4d0>) of role type named set2prop_type
% Using role type
% Declaring set2prop:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x9d2f80>, <kernel.Sort object at 0xc80f38>) of role type named prop2set2propI_type
% Using role type
% Declaring prop2set2propI:Prop
% FOF formula (((eq Prop) prop2set2propI) (forall (Xphi:Prop), (Xphi->(set2prop (prop2set Xphi))))) of role definition named prop2set2propI
% A new definition: (((eq Prop) prop2set2propI) (forall (Xphi:Prop), (Xphi->(set2prop (prop2set Xphi)))))
% Defined: prop2set2propI:=(forall (Xphi:Prop), (Xphi->(set2prop (prop2set Xphi))))
% FOF formula (<kernel.Constant object at 0x9d2dd0>, <kernel.Sort object at 0xc80f38>) of role type named notdexE_type
% Using role type
% Declaring notdexE:Prop
% FOF formula (((eq Prop) notdexE) (forall (A:fofType) (Xphi:(fofType->Prop)), ((((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->False)->(forall (Xx:fofType), (((in Xx) A)->((Xphi Xx)->False)))))) of role definition named notdexE
% A new definition: (((eq Prop) notdexE) (forall (A:fofType) (Xphi:(fofType->Prop)), ((((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->False)->(forall (Xx:fofType), (((in Xx) A)->((Xphi Xx)->False))))))
% Defined: notdexE:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->False)->(forall (Xx:fofType), (((in Xx) A)->((Xphi Xx)->False)))))
% FOF formula (<kernel.Constant object at 0xbbe050>, <kernel.Sort object at 0xc80f38>) of role type named notdallE_type
% Using role type
% Declaring notdallE:Prop
% FOF formula (((eq Prop) notdallE) (forall (A:fofType) (Xphi:(fofType->Prop)), (((forall (Xx:fofType), (((in Xx) A)->(Xphi Xx)))->False)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((Xphi Xx)->False))))))) of role definition named notdallE
% A new definition: (((eq Prop) notdallE) (forall (A:fofType) (Xphi:(fofType->Prop)), (((forall (Xx:fofType), (((in Xx) A)->(Xphi Xx)))->False)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((Xphi Xx)->False)))))))
% Defined: notdallE:=(forall (A:fofType) (Xphi:(fofType->Prop)), (((forall (Xx:fofType), (((in Xx) A)->(Xphi Xx)))->False)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((Xphi Xx)->False))))))
% FOF formula (<kernel.Constant object at 0xbbe488>, <kernel.Sort object at 0xc80f38>) of role type named exuI1_type
% Using role type
% Declaring exuI1:Prop
% FOF formula (((eq Prop) exuI1) (forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(exu (fun (Xx:fofType)=> (Xphi Xx)))))) of role definition named exuI1
% A new definition: (((eq Prop) exuI1) (forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(exu (fun (Xx:fofType)=> (Xphi Xx))))))
% Defined: exuI1:=(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(exu (fun (Xx:fofType)=> (Xphi Xx)))))
% FOF formula (<kernel.Constant object at 0xbbe8c0>, <kernel.Sort object at 0xc80f38>) of role type named exuI3_type
% Using role type
% Declaring exuI3:Prop
% FOF formula (((eq Prop) exuI3) (forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (Xphi Xx)))->((forall (Xx:fofType) (Xy:fofType), ((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy))))->(exu (fun (Xx:fofType)=> (Xphi Xx))))))) of role definition named exuI3
% A new definition: (((eq Prop) exuI3) (forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (Xphi Xx)))->((forall (Xx:fofType) (Xy:fofType), ((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy))))->(exu (fun (Xx:fofType)=> (Xphi Xx)))))))
% Defined: exuI3:=(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (Xphi Xx)))->((forall (Xx:fofType) (Xy:fofType), ((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy))))->(exu (fun (Xx:fofType)=> (Xphi Xx))))))
% FOF formula (<kernel.Constant object at 0xbbe320>, <kernel.Sort object at 0xc80f38>) of role type named exuI2_type
% Using role type
% Declaring exuI2:Prop
% FOF formula (((eq Prop) exuI2) (forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(exu (fun (Xx:fofType)=> (Xphi Xx)))))) of role definition named exuI2
% A new definition: (((eq Prop) exuI2) (forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(exu (fun (Xx:fofType)=> (Xphi Xx))))))
% Defined: exuI2:=(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(exu (fun (Xx:fofType)=> (Xphi Xx)))))
% FOF formula (<kernel.Constant object at 0xbbe7a0>, <kernel.Sort object at 0xc80f38>) of role type named inCongP_type
% Using role type
% Declaring inCongP:Prop
% FOF formula (((eq Prop) inCongP) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(((in Xx) A)->((in Xy) B))))))) of role definition named inCongP
% A new definition: (((eq Prop) inCongP) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(((in Xx) A)->((in Xy) B)))))))
% Defined: inCongP:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(((in Xx) A)->((in Xy) B))))))
% FOF formula (<kernel.Constant object at 0xbbe830>, <kernel.Sort object at 0xc80f38>) of role type named in__Cong_type
% Using role type
% Declaring in__Cong:Prop
% FOF formula (((eq Prop) in__Cong) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((iff ((in Xx) A)) ((in Xy) B))))))) of role definition named in__Cong
% A new definition: (((eq Prop) in__Cong) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((iff ((in Xx) A)) ((in Xy) B)))))))
% Defined: in__Cong:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((iff ((in Xx) A)) ((in Xy) B))))))
% FOF formula (<kernel.Constant object at 0xbbe7e8>, <kernel.Sort object at 0xc80f38>) of role type named exuE3u_type
% Using role type
% Declaring exuE3u:Prop
% FOF formula (((eq Prop) exuE3u) (forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->(forall (Xx:fofType) (Xy:fofType), ((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy))))))) of role definition named exuE3u
% A new definition: (((eq Prop) exuE3u) (forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->(forall (Xx:fofType) (Xy:fofType), ((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))))
% Defined: exuE3u:=(forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->(forall (Xx:fofType) (Xy:fofType), ((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% FOF formula (<kernel.Constant object at 0xbbe560>, <kernel.Sort object at 0xc80f38>) of role type named exu__Cong_type
% Using role type
% Declaring exu__Cong:Prop
% FOF formula (((eq Prop) exu__Cong) (forall (Xphi:(fofType->Prop)) (Xpsi:(fofType->Prop)), ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((iff (Xphi Xx)) (Xpsi Xy))))->((iff (exu (fun (Xx:fofType)=> (Xphi Xx)))) (exu (fun (Xx:fofType)=> (Xpsi Xx))))))) of role definition named exu__Cong
% A new definition: (((eq Prop) exu__Cong) (forall (Xphi:(fofType->Prop)) (Xpsi:(fofType->Prop)), ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((iff (Xphi Xx)) (Xpsi Xy))))->((iff (exu (fun (Xx:fofType)=> (Xphi Xx)))) (exu (fun (Xx:fofType)=> (Xpsi Xx)))))))
% Defined: exu__Cong:=(forall (Xphi:(fofType->Prop)) (Xpsi:(fofType->Prop)), ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((iff (Xphi Xx)) (Xpsi Xy))))->((iff (exu (fun (Xx:fofType)=> (Xphi Xx)))) (exu (fun (Xx:fofType)=> (Xpsi Xx))))))
% FOF formula (<kernel.Constant object at 0xbbe638>, <kernel.Sort object at 0xc80f38>) of role type named emptyset__Cong_type
% Using role type
% Declaring emptyset__Cong:Prop
% FOF formula (((eq Prop) emptyset__Cong) (((eq fofType) emptyset) emptyset)) of role definition named emptyset__Cong
% A new definition: (((eq Prop) emptyset__Cong) (((eq fofType) emptyset) emptyset))
% Defined: emptyset__Cong:=(((eq fofType) emptyset) emptyset)
% FOF formula (<kernel.Constant object at 0xbbecf8>, <kernel.Sort object at 0xc80f38>) of role type named setadjoin__Cong_type
% Using role type
% Declaring setadjoin__Cong:Prop
% FOF formula (((eq Prop) setadjoin__Cong) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(forall (Xz:fofType) (Xu:fofType), ((((eq fofType) Xz) Xu)->(((eq fofType) ((setadjoin Xx) Xz)) ((setadjoin Xy) Xu))))))) of role definition named setadjoin__Cong
% A new definition: (((eq Prop) setadjoin__Cong) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(forall (Xz:fofType) (Xu:fofType), ((((eq fofType) Xz) Xu)->(((eq fofType) ((setadjoin Xx) Xz)) ((setadjoin Xy) Xu)))))))
% Defined: setadjoin__Cong:=(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(forall (Xz:fofType) (Xu:fofType), ((((eq fofType) Xz) Xu)->(((eq fofType) ((setadjoin Xx) Xz)) ((setadjoin Xy) Xu))))))
% FOF formula (<kernel.Constant object at 0xbbee18>, <kernel.Sort object at 0xc80f38>) of role type named powerset__Cong_type
% Using role type
% Declaring powerset__Cong:Prop
% FOF formula (((eq Prop) powerset__Cong) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(((eq fofType) (powerset A)) (powerset B))))) of role definition named powerset__Cong
% A new definition: (((eq Prop) powerset__Cong) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(((eq fofType) (powerset A)) (powerset B)))))
% Defined: powerset__Cong:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(((eq fofType) (powerset A)) (powerset B))))
% FOF formula (<kernel.Constant object at 0xbbecb0>, <kernel.Sort object at 0xc80f38>) of role type named setunion__Cong_type
% Using role type
% Declaring setunion__Cong:Prop
% FOF formula (((eq Prop) setunion__Cong) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(((eq fofType) (setunion A)) (setunion B))))) of role definition named setunion__Cong
% A new definition: (((eq Prop) setunion__Cong) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(((eq fofType) (setunion A)) (setunion B)))))
% Defined: setunion__Cong:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(((eq fofType) (setunion A)) (setunion B))))
% FOF formula (<kernel.Constant object at 0xbbef38>, <kernel.Sort object at 0xc80f38>) of role type named omega__Cong_type
% Using role type
% Declaring omega__Cong:Prop
% FOF formula (((eq Prop) omega__Cong) (((eq fofType) omega) omega)) of role definition named omega__Cong
% A new definition: (((eq Prop) omega__Cong) (((eq fofType) omega) omega))
% Defined: omega__Cong:=(((eq fofType) omega) omega)
% FOF formula (<kernel.Constant object at 0xbbec20>, <kernel.Sort object at 0xc80f38>) of role type named exuEu_type
% Using role type
% Declaring exuEu:Prop
% FOF formula (((eq Prop) exuEu) (forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->(forall (Xx:fofType) (Xy:fofType), ((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy))))))) of role definition named exuEu
% A new definition: (((eq Prop) exuEu) (forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->(forall (Xx:fofType) (Xy:fofType), ((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))))
% Defined: exuEu:=(forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->(forall (Xx:fofType) (Xy:fofType), ((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy))))))
% FOF formula (<kernel.Constant object at 0xbbea70>, <kernel.Sort object at 0xc80f38>) of role type named descr__Cong_type
% Using role type
% Declaring descr__Cong:Prop
% FOF formula (((eq Prop) descr__Cong) (forall (Xphi:(fofType->Prop)) (Xpsi:(fofType->Prop)), ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((iff (Xphi Xx)) (Xpsi Xy))))->((exu (fun (Xx:fofType)=> (Xphi Xx)))->((exu (fun (Xx:fofType)=> (Xpsi Xx)))->(((eq fofType) (descr (fun (Xx:fofType)=> (Xphi Xx)))) (descr (fun (Xx:fofType)=> (Xpsi Xx))))))))) of role definition named descr__Cong
% A new definition: (((eq Prop) descr__Cong) (forall (Xphi:(fofType->Prop)) (Xpsi:(fofType->Prop)), ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((iff (Xphi Xx)) (Xpsi Xy))))->((exu (fun (Xx:fofType)=> (Xphi Xx)))->((exu (fun (Xx:fofType)=> (Xpsi Xx)))->(((eq fofType) (descr (fun (Xx:fofType)=> (Xphi Xx)))) (descr (fun (Xx:fofType)=> (Xpsi Xx)))))))))
% Defined: descr__Cong:=(forall (Xphi:(fofType->Prop)) (Xpsi:(fofType->Prop)), ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((iff (Xphi Xx)) (Xpsi Xy))))->((exu (fun (Xx:fofType)=> (Xphi Xx)))->((exu (fun (Xx:fofType)=> (Xpsi Xx)))->(((eq fofType) (descr (fun (Xx:fofType)=> (Xphi Xx)))) (descr (fun (Xx:fofType)=> (Xpsi Xx))))))))
% FOF formula (<kernel.Constant object at 0xbbe908>, <kernel.Sort object at 0xc80f38>) of role type named dsetconstr__Cong_type
% Using role type
% Declaring dsetconstr__Cong:Prop
% FOF formula (((eq Prop) dsetconstr__Cong) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(forall (Xphi:(fofType->Prop)) (Xpsi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((((eq fofType) Xx) Xy)->((iff (Xphi Xx)) (Xpsi Xy)))))))->(((eq fofType) ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) ((dsetconstr B) (fun (Xx:fofType)=> (Xpsi Xx))))))))) of role definition named dsetconstr__Cong
% A new definition: (((eq Prop) dsetconstr__Cong) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(forall (Xphi:(fofType->Prop)) (Xpsi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((((eq fofType) Xx) Xy)->((iff (Xphi Xx)) (Xpsi Xy)))))))->(((eq fofType) ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) ((dsetconstr B) (fun (Xx:fofType)=> (Xpsi Xx)))))))))
% Defined: dsetconstr__Cong:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(forall (Xphi:(fofType->Prop)) (Xpsi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((((eq fofType) Xx) Xy)->((iff (Xphi Xx)) (Xpsi Xy)))))))->(((eq fofType) ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) ((dsetconstr B) (fun (Xx:fofType)=> (Xpsi Xx))))))))
% FOF formula (<kernel.Constant object at 0xbbeab8>, <kernel.DependentProduct object at 0xbbefc8>) of role type named subset_type
% Using role type
% Declaring subset:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xbbe9e0>, <kernel.DependentProduct object at 0xbbeea8>) of role type named disjoint_type
% Using role type
% Declaring disjoint:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xbbe908>, <kernel.DependentProduct object at 0xbbeab8>) of role type named setsmeet_type
% Using role type
% Declaring setsmeet:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xbbedd0>, <kernel.Sort object at 0xc80f38>) of role type named subsetI1_type
% Using role type
% Declaring subsetI1:Prop
% FOF formula (((eq Prop) subsetI1) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((subset A) B)))) of role definition named subsetI1
% A new definition: (((eq Prop) subsetI1) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((subset A) B))))
% Defined: subsetI1:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((subset A) B)))
% FOF formula (<kernel.Constant object at 0xbbed40>, <kernel.Sort object at 0xc80f38>) of role type named eqimpsubset2_type
% Using role type
% Declaring eqimpsubset2:Prop
% FOF formula (((eq Prop) eqimpsubset2) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->((subset B) A)))) of role definition named eqimpsubset2
% A new definition: (((eq Prop) eqimpsubset2) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->((subset B) A))))
% Defined: eqimpsubset2:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->((subset B) A)))
% FOF formula (<kernel.Constant object at 0xbbe908>, <kernel.Sort object at 0xc80f38>) of role type named eqimpsubset1_type
% Using role type
% Declaring eqimpsubset1:Prop
% FOF formula (((eq Prop) eqimpsubset1) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->((subset A) B)))) of role definition named eqimpsubset1
% A new definition: (((eq Prop) eqimpsubset1) (forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->((subset A) B))))
% Defined: eqimpsubset1:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->((subset A) B)))
% FOF formula (<kernel.Constant object at 0xbbeea8>, <kernel.Sort object at 0xc80f38>) of role type named subsetI2_type
% Using role type
% Declaring subsetI2:Prop
% FOF formula (((eq Prop) subsetI2) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((subset A) B)))) of role definition named subsetI2
% A new definition: (((eq Prop) subsetI2) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((subset A) B))))
% Defined: subsetI2:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((subset A) B)))
% FOF formula (<kernel.Constant object at 0xbbe6c8>, <kernel.Sort object at 0xc80f38>) of role type named emptysetsubset_type
% Using role type
% Declaring emptysetsubset:Prop
% FOF formula (((eq Prop) emptysetsubset) (forall (A:fofType), ((subset emptyset) A))) of role definition named emptysetsubset
% A new definition: (((eq Prop) emptysetsubset) (forall (A:fofType), ((subset emptyset) A)))
% Defined: emptysetsubset:=(forall (A:fofType), ((subset emptyset) A))
% FOF formula (<kernel.Constant object at 0xbbe1b8>, <kernel.Sort object at 0xc80f38>) of role type named subsetE_type
% Using role type
% Declaring subsetE:Prop
% FOF formula (((eq Prop) subsetE) (forall (A:fofType) (B:fofType) (Xx:fofType), (((subset A) B)->(((in Xx) A)->((in Xx) B))))) of role definition named subsetE
% A new definition: (((eq Prop) subsetE) (forall (A:fofType) (B:fofType) (Xx:fofType), (((subset A) B)->(((in Xx) A)->((in Xx) B)))))
% Defined: subsetE:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((subset A) B)->(((in Xx) A)->((in Xx) B))))
% FOF formula (<kernel.Constant object at 0xbbe0e0>, <kernel.Sort object at 0xc80f38>) of role type named subsetE2_type
% Using role type
% Declaring subsetE2:Prop
% FOF formula (((eq Prop) subsetE2) (forall (A:fofType) (B:fofType) (Xx:fofType), (((subset A) B)->((((in Xx) B)->False)->(((in Xx) A)->False))))) of role definition named subsetE2
% A new definition: (((eq Prop) subsetE2) (forall (A:fofType) (B:fofType) (Xx:fofType), (((subset A) B)->((((in Xx) B)->False)->(((in Xx) A)->False)))))
% Defined: subsetE2:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((subset A) B)->((((in Xx) B)->False)->(((in Xx) A)->False))))
% FOF formula (<kernel.Constant object at 0xbbe518>, <kernel.Sort object at 0xc80f38>) of role type named notsubsetI_type
% Using role type
% Declaring notsubsetI:Prop
% FOF formula (((eq Prop) notsubsetI) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->(((subset A) B)->False))))) of role definition named notsubsetI
% A new definition: (((eq Prop) notsubsetI) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->(((subset A) B)->False)))))
% Defined: notsubsetI:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->(((subset A) B)->False))))
% FOF formula (<kernel.Constant object at 0xbbe518>, <kernel.Sort object at 0xc80f38>) of role type named notequalI1_type
% Using role type
% Declaring notequalI1:Prop
% FOF formula (((eq Prop) notequalI1) (forall (A:fofType) (B:fofType), ((((subset A) B)->False)->(not (((eq fofType) A) B))))) of role definition named notequalI1
% A new definition: (((eq Prop) notequalI1) (forall (A:fofType) (B:fofType), ((((subset A) B)->False)->(not (((eq fofType) A) B)))))
% Defined: notequalI1:=(forall (A:fofType) (B:fofType), ((((subset A) B)->False)->(not (((eq fofType) A) B))))
% FOF formula (<kernel.Constant object at 0xbbe518>, <kernel.Sort object at 0xc80f38>) of role type named notequalI2_type
% Using role type
% Declaring notequalI2:Prop
% FOF formula (((eq Prop) notequalI2) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->(not (((eq fofType) A) B)))))) of role definition named notequalI2
% A new definition: (((eq Prop) notequalI2) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->(not (((eq fofType) A) B))))))
% Defined: notequalI2:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->(not (((eq fofType) A) B)))))
% FOF formula (<kernel.Constant object at 0xbc1368>, <kernel.Sort object at 0xc80f38>) of role type named subsetRefl_type
% Using role type
% Declaring subsetRefl:Prop
% FOF formula (((eq Prop) subsetRefl) (forall (A:fofType), ((subset A) A))) of role definition named subsetRefl
% A new definition: (((eq Prop) subsetRefl) (forall (A:fofType), ((subset A) A)))
% Defined: subsetRefl:=(forall (A:fofType), ((subset A) A))
% FOF formula (<kernel.Constant object at 0xbc1710>, <kernel.Sort object at 0xc80f38>) of role type named subsetTrans_type
% Using role type
% Declaring subsetTrans:Prop
% FOF formula (((eq Prop) subsetTrans) (forall (A:fofType) (B:fofType) (C:fofType), (((subset A) B)->(((subset B) C)->((subset A) C))))) of role definition named subsetTrans
% A new definition: (((eq Prop) subsetTrans) (forall (A:fofType) (B:fofType) (C:fofType), (((subset A) B)->(((subset B) C)->((subset A) C)))))
% Defined: subsetTrans:=(forall (A:fofType) (B:fofType) (C:fofType), (((subset A) B)->(((subset B) C)->((subset A) C))))
% FOF formula (<kernel.Constant object at 0xbc13b0>, <kernel.Sort object at 0xc80f38>) of role type named setadjoinSub_type
% Using role type
% Declaring setadjoinSub:Prop
% FOF formula (((eq Prop) setadjoinSub) (forall (Xx:fofType) (A:fofType), ((subset A) ((setadjoin Xx) A)))) of role definition named setadjoinSub
% A new definition: (((eq Prop) setadjoinSub) (forall (Xx:fofType) (A:fofType), ((subset A) ((setadjoin Xx) A))))
% Defined: setadjoinSub:=(forall (Xx:fofType) (A:fofType), ((subset A) ((setadjoin Xx) A)))
% FOF formula (<kernel.Constant object at 0xbc15a8>, <kernel.Sort object at 0xc80f38>) of role type named setadjoinSub2_type
% Using role type
% Declaring setadjoinSub2:Prop
% FOF formula (((eq Prop) setadjoinSub2) (forall (A:fofType) (Xx:fofType) (B:fofType), (((subset A) B)->((subset A) ((setadjoin Xx) B))))) of role definition named setadjoinSub2
% A new definition: (((eq Prop) setadjoinSub2) (forall (A:fofType) (Xx:fofType) (B:fofType), (((subset A) B)->((subset A) ((setadjoin Xx) B)))))
% Defined: setadjoinSub2:=(forall (A:fofType) (Xx:fofType) (B:fofType), (((subset A) B)->((subset A) ((setadjoin Xx) B))))
% FOF formula (<kernel.Constant object at 0xbc1128>, <kernel.Sort object at 0xc80f38>) of role type named subset2powerset_type
% Using role type
% Declaring subset2powerset:Prop
% FOF formula (((eq Prop) subset2powerset) (forall (A:fofType) (B:fofType), (((subset A) B)->((in A) (powerset B))))) of role definition named subset2powerset
% A new definition: (((eq Prop) subset2powerset) (forall (A:fofType) (B:fofType), (((subset A) B)->((in A) (powerset B)))))
% Defined: subset2powerset:=(forall (A:fofType) (B:fofType), (((subset A) B)->((in A) (powerset B))))
% FOF formula (<kernel.Constant object at 0xbc18c0>, <kernel.Sort object at 0xc80f38>) of role type named setextsub_type
% Using role type
% Declaring setextsub:Prop
% FOF formula (((eq Prop) setextsub) (forall (A:fofType) (B:fofType), (((subset A) B)->(((subset B) A)->(((eq fofType) A) B))))) of role definition named setextsub
% A new definition: (((eq Prop) setextsub) (forall (A:fofType) (B:fofType), (((subset A) B)->(((subset B) A)->(((eq fofType) A) B)))))
% Defined: setextsub:=(forall (A:fofType) (B:fofType), (((subset A) B)->(((subset B) A)->(((eq fofType) A) B))))
% FOF formula (<kernel.Constant object at 0xbc1680>, <kernel.Sort object at 0xc80f38>) of role type named subsetemptysetimpeq_type
% Using role type
% Declaring subsetemptysetimpeq:Prop
% FOF formula (((eq Prop) subsetemptysetimpeq) (forall (A:fofType), (((subset A) emptyset)->(((eq fofType) A) emptyset)))) of role definition named subsetemptysetimpeq
% A new definition: (((eq Prop) subsetemptysetimpeq) (forall (A:fofType), (((subset A) emptyset)->(((eq fofType) A) emptyset))))
% Defined: subsetemptysetimpeq:=(forall (A:fofType), (((subset A) emptyset)->(((eq fofType) A) emptyset)))
% FOF formula (<kernel.Constant object at 0xbc1200>, <kernel.Sort object at 0xc80f38>) of role type named powersetI1_type
% Using role type
% Declaring powersetI1:Prop
% FOF formula (((eq Prop) powersetI1) (forall (A:fofType) (B:fofType), (((subset B) A)->((in B) (powerset A))))) of role definition named powersetI1
% A new definition: (((eq Prop) powersetI1) (forall (A:fofType) (B:fofType), (((subset B) A)->((in B) (powerset A)))))
% Defined: powersetI1:=(forall (A:fofType) (B:fofType), (((subset B) A)->((in B) (powerset A))))
% FOF formula (<kernel.Constant object at 0xbc1950>, <kernel.Sort object at 0xc80f38>) of role type named powersetE1_type
% Using role type
% Declaring powersetE1:Prop
% FOF formula (((eq Prop) powersetE1) (forall (A:fofType) (B:fofType), (((in B) (powerset A))->((subset B) A)))) of role definition named powersetE1
% A new definition: (((eq Prop) powersetE1) (forall (A:fofType) (B:fofType), (((in B) (powerset A))->((subset B) A))))
% Defined: powersetE1:=(forall (A:fofType) (B:fofType), (((in B) (powerset A))->((subset B) A)))
% FOF formula (<kernel.Constant object at 0xbc1638>, <kernel.Sort object at 0xc80f38>) of role type named inPowerset_type
% Using role type
% Declaring inPowerset:Prop
% FOF formula (((eq Prop) inPowerset) (forall (A:fofType), ((in A) (powerset A)))) of role definition named inPowerset
% A new definition: (((eq Prop) inPowerset) (forall (A:fofType), ((in A) (powerset A))))
% Defined: inPowerset:=(forall (A:fofType), ((in A) (powerset A)))
% FOF formula (<kernel.Constant object at 0xbc10e0>, <kernel.Sort object at 0xc80f38>) of role type named powersetsubset_type
% Using role type
% Declaring powersetsubset:Prop
% FOF formula (((eq Prop) powersetsubset) (forall (A:fofType) (B:fofType), (((subset A) B)->((subset (powerset A)) (powerset B))))) of role definition named powersetsubset
% A new definition: (((eq Prop) powersetsubset) (forall (A:fofType) (B:fofType), (((subset A) B)->((subset (powerset A)) (powerset B)))))
% Defined: powersetsubset:=(forall (A:fofType) (B:fofType), (((subset A) B)->((subset (powerset A)) (powerset B))))
% FOF formula (<kernel.Constant object at 0xbc14d0>, <kernel.Sort object at 0xc80f38>) of role type named sepInPowerset_type
% Using role type
% Declaring sepInPowerset:Prop
% FOF formula (((eq Prop) sepInPowerset) (forall (A:fofType) (Xphi:(fofType->Prop)), ((in ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) (powerset A)))) of role definition named sepInPowerset
% A new definition: (((eq Prop) sepInPowerset) (forall (A:fofType) (Xphi:(fofType->Prop)), ((in ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) (powerset A))))
% Defined: sepInPowerset:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((in ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) (powerset A)))
% FOF formula (<kernel.Constant object at 0xbc1830>, <kernel.Sort object at 0xc80f38>) of role type named sepSubset_type
% Using role type
% Declaring sepSubset:Prop
% FOF formula (((eq Prop) sepSubset) (forall (A:fofType) (Xphi:(fofType->Prop)), ((subset ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) A))) of role definition named sepSubset
% A new definition: (((eq Prop) sepSubset) (forall (A:fofType) (Xphi:(fofType->Prop)), ((subset ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) A)))
% Defined: sepSubset:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((subset ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) A))
% FOF formula (<kernel.Constant object at 0xbc1b00>, <kernel.DependentProduct object at 0xbc17a0>) of role type named binunion_type
% Using role type
% Declaring binunion:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0xbc1518>, <kernel.Sort object at 0xc80f38>) of role type named binunionIL_type
% Using role type
% Declaring binunionIL:Prop
% FOF formula (((eq Prop) binunionIL) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((in Xx) ((binunion A) B))))) of role definition named binunionIL
% A new definition: (((eq Prop) binunionIL) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((in Xx) ((binunion A) B)))))
% Defined: binunionIL:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((in Xx) ((binunion A) B))))
% FOF formula (<kernel.Constant object at 0xbc1830>, <kernel.Sort object at 0xc80f38>) of role type named upairset2IR_type
% Using role type
% Declaring upairset2IR:Prop
% FOF formula (((eq Prop) upairset2IR) (forall (Xx:fofType) (Xy:fofType), ((in Xy) ((setadjoin Xx) ((setadjoin Xy) emptyset))))) of role definition named upairset2IR
% A new definition: (((eq Prop) upairset2IR) (forall (Xx:fofType) (Xy:fofType), ((in Xy) ((setadjoin Xx) ((setadjoin Xy) emptyset)))))
% Defined: upairset2IR:=(forall (Xx:fofType) (Xy:fofType), ((in Xy) ((setadjoin Xx) ((setadjoin Xy) emptyset))))
% FOF formula (<kernel.Constant object at 0xbc1ef0>, <kernel.Sort object at 0xc80f38>) of role type named binunionIR_type
% Using role type
% Declaring binunionIR:Prop
% FOF formula (((eq Prop) binunionIR) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) B)->((in Xx) ((binunion A) B))))) of role definition named binunionIR
% A new definition: (((eq Prop) binunionIR) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) B)->((in Xx) ((binunion A) B)))))
% Defined: binunionIR:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) B)->((in Xx) ((binunion A) B))))
% FOF formula (<kernel.Constant object at 0xbc1ea8>, <kernel.Sort object at 0xc80f38>) of role type named binunionEcases_type
% Using role type
% Declaring binunionEcases:Prop
% FOF formula (((eq Prop) binunionEcases) (forall (A:fofType) (B:fofType) (Xx:fofType) (Xphi:Prop), (((in Xx) ((binunion A) B))->((((in Xx) A)->Xphi)->((((in Xx) B)->Xphi)->Xphi))))) of role definition named binunionEcases
% A new definition: (((eq Prop) binunionEcases) (forall (A:fofType) (B:fofType) (Xx:fofType) (Xphi:Prop), (((in Xx) ((binunion A) B))->((((in Xx) A)->Xphi)->((((in Xx) B)->Xphi)->Xphi)))))
% Defined: binunionEcases:=(forall (A:fofType) (B:fofType) (Xx:fofType) (Xphi:Prop), (((in Xx) ((binunion A) B))->((((in Xx) A)->Xphi)->((((in Xx) B)->Xphi)->Xphi))))
% FOF formula (<kernel.Constant object at 0xbc15f0>, <kernel.Sort object at 0xc80f38>) of role type named binunionE_type
% Using role type
% Declaring binunionE:Prop
% FOF formula (((eq Prop) binunionE) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((binunion A) B))->((or ((in Xx) A)) ((in Xx) B))))) of role definition named binunionE
% A new definition: (((eq Prop) binunionE) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((binunion A) B))->((or ((in Xx) A)) ((in Xx) B)))))
% Defined: binunionE:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((binunion A) B))->((or ((in Xx) A)) ((in Xx) B))))
% FOF formula (<kernel.Constant object at 0xbc1e60>, <kernel.Sort object at 0xc80f38>) of role type named binunionLsub_type
% Using role type
% Declaring binunionLsub:Prop
% FOF formula (((eq Prop) binunionLsub) (forall (A:fofType) (B:fofType), ((subset A) ((binunion A) B)))) of role definition named binunionLsub
% A new definition: (((eq Prop) binunionLsub) (forall (A:fofType) (B:fofType), ((subset A) ((binunion A) B))))
% Defined: binunionLsub:=(forall (A:fofType) (B:fofType), ((subset A) ((binunion A) B)))
% FOF formula (<kernel.Constant object at 0xbc1b90>, <kernel.Sort object at 0xc80f38>) of role type named binunionRsub_type
% Using role type
% Declaring binunionRsub:Prop
% FOF formula (((eq Prop) binunionRsub) (forall (A:fofType) (B:fofType), ((subset B) ((binunion A) B)))) of role definition named binunionRsub
% A new definition: (((eq Prop) binunionRsub) (forall (A:fofType) (B:fofType), ((subset B) ((binunion A) B))))
% Defined: binunionRsub:=(forall (A:fofType) (B:fofType), ((subset B) ((binunion A) B)))
% FOF formula (<kernel.Constant object at 0xbc1cf8>, <kernel.DependentProduct object at 0xbc1f38>) of role type named binintersect_type
% Using role type
% Declaring binintersect:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0xbc1cb0>, <kernel.Sort object at 0xc80f38>) of role type named binintersectI_type
% Using role type
% Declaring binintersectI:Prop
% FOF formula (((eq Prop) binintersectI) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(((in Xx) B)->((in Xx) ((binintersect A) B)))))) of role definition named binintersectI
% A new definition: (((eq Prop) binintersectI) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(((in Xx) B)->((in Xx) ((binintersect A) B))))))
% Defined: binintersectI:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(((in Xx) B)->((in Xx) ((binintersect A) B)))))
% FOF formula (<kernel.Constant object at 0xbc1b90>, <kernel.Sort object at 0xc80f38>) of role type named binintersectSubset5_type
% Using role type
% Declaring binintersectSubset5:Prop
% FOF formula (((eq Prop) binintersectSubset5) (forall (A:fofType) (B:fofType) (C:fofType), (((subset C) A)->(((subset C) B)->((subset C) ((binintersect A) B)))))) of role definition named binintersectSubset5
% A new definition: (((eq Prop) binintersectSubset5) (forall (A:fofType) (B:fofType) (C:fofType), (((subset C) A)->(((subset C) B)->((subset C) ((binintersect A) B))))))
% Defined: binintersectSubset5:=(forall (A:fofType) (B:fofType) (C:fofType), (((subset C) A)->(((subset C) B)->((subset C) ((binintersect A) B)))))
% FOF formula (<kernel.Constant object at 0xbc1cf8>, <kernel.Sort object at 0xc80f38>) of role type named binintersectEL_type
% Using role type
% Declaring binintersectEL:Prop
% FOF formula (((eq Prop) binintersectEL) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((binintersect A) B))->((in Xx) A)))) of role definition named binintersectEL
% A new definition: (((eq Prop) binintersectEL) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((binintersect A) B))->((in Xx) A))))
% Defined: binintersectEL:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((binintersect A) B))->((in Xx) A)))
% FOF formula (<kernel.Constant object at 0xbc1b48>, <kernel.Sort object at 0xc80f38>) of role type named binintersectLsub_type
% Using role type
% Declaring binintersectLsub:Prop
% FOF formula (((eq Prop) binintersectLsub) (forall (A:fofType) (B:fofType), ((subset ((binintersect A) B)) A))) of role definition named binintersectLsub
% A new definition: (((eq Prop) binintersectLsub) (forall (A:fofType) (B:fofType), ((subset ((binintersect A) B)) A)))
% Defined: binintersectLsub:=(forall (A:fofType) (B:fofType), ((subset ((binintersect A) B)) A))
% FOF formula (<kernel.Constant object at 0xbc1d40>, <kernel.Sort object at 0xc80f38>) of role type named binintersectSubset2_type
% Using role type
% Declaring binintersectSubset2:Prop
% FOF formula (((eq Prop) binintersectSubset2) (forall (A:fofType) (B:fofType), (((subset A) B)->(((eq fofType) ((binintersect A) B)) A)))) of role definition named binintersectSubset2
% A new definition: (((eq Prop) binintersectSubset2) (forall (A:fofType) (B:fofType), (((subset A) B)->(((eq fofType) ((binintersect A) B)) A))))
% Defined: binintersectSubset2:=(forall (A:fofType) (B:fofType), (((subset A) B)->(((eq fofType) ((binintersect A) B)) A)))
% FOF formula (<kernel.Constant object at 0xbc1d40>, <kernel.Sort object at 0xc80f38>) of role type named binintersectSubset3_type
% Using role type
% Declaring binintersectSubset3:Prop
% FOF formula (((eq Prop) binintersectSubset3) (forall (A:fofType) (B:fofType), ((((eq fofType) ((binintersect A) B)) B)->((subset B) A)))) of role definition named binintersectSubset3
% A new definition: (((eq Prop) binintersectSubset3) (forall (A:fofType) (B:fofType), ((((eq fofType) ((binintersect A) B)) B)->((subset B) A))))
% Defined: binintersectSubset3:=(forall (A:fofType) (B:fofType), ((((eq fofType) ((binintersect A) B)) B)->((subset B) A)))
% FOF formula (<kernel.Constant object at 0xbc1d40>, <kernel.Sort object at 0xc80f38>) of role type named binintersectER_type
% Using role type
% Declaring binintersectER:Prop
% FOF formula (((eq Prop) binintersectER) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((binintersect A) B))->((in Xx) B)))) of role definition named binintersectER
% A new definition: (((eq Prop) binintersectER) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((binintersect A) B))->((in Xx) B))))
% Defined: binintersectER:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((binintersect A) B))->((in Xx) B)))
% FOF formula (<kernel.Constant object at 0xbc95a8>, <kernel.Sort object at 0xc80f38>) of role type named disjointsetsI1_type
% Using role type
% Declaring disjointsetsI1:Prop
% FOF formula (((eq Prop) disjointsetsI1) (forall (A:fofType) (B:fofType), ((((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((in Xx) B))))->False)->(((eq fofType) ((binintersect A) B)) emptyset)))) of role definition named disjointsetsI1
% A new definition: (((eq Prop) disjointsetsI1) (forall (A:fofType) (B:fofType), ((((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((in Xx) B))))->False)->(((eq fofType) ((binintersect A) B)) emptyset))))
% Defined: disjointsetsI1:=(forall (A:fofType) (B:fofType), ((((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((in Xx) B))))->False)->(((eq fofType) ((binintersect A) B)) emptyset)))
% FOF formula (<kernel.Constant object at 0xbc9440>, <kernel.Sort object at 0xc80f38>) of role type named binintersectRsub_type
% Using role type
% Declaring binintersectRsub:Prop
% FOF formula (((eq Prop) binintersectRsub) (forall (A:fofType) (B:fofType), ((subset ((binintersect A) B)) B))) of role definition named binintersectRsub
% A new definition: (((eq Prop) binintersectRsub) (forall (A:fofType) (B:fofType), ((subset ((binintersect A) B)) B)))
% Defined: binintersectRsub:=(forall (A:fofType) (B:fofType), ((subset ((binintersect A) B)) B))
% FOF formula (<kernel.Constant object at 0xbc9050>, <kernel.Sort object at 0xc80f38>) of role type named binintersectSubset4_type
% Using role type
% Declaring binintersectSubset4:Prop
% FOF formula (((eq Prop) binintersectSubset4) (forall (A:fofType) (B:fofType), (((subset B) A)->(((eq fofType) ((binintersect A) B)) B)))) of role definition named binintersectSubset4
% A new definition: (((eq Prop) binintersectSubset4) (forall (A:fofType) (B:fofType), (((subset B) A)->(((eq fofType) ((binintersect A) B)) B))))
% Defined: binintersectSubset4:=(forall (A:fofType) (B:fofType), (((subset B) A)->(((eq fofType) ((binintersect A) B)) B)))
% FOF formula (<kernel.Constant object at 0xbc9488>, <kernel.Sort object at 0xc80f38>) of role type named binintersectSubset1_type
% Using role type
% Declaring binintersectSubset1:Prop
% FOF formula (((eq Prop) binintersectSubset1) (forall (A:fofType) (B:fofType), ((((eq fofType) ((binintersect A) B)) A)->((subset A) B)))) of role definition named binintersectSubset1
% A new definition: (((eq Prop) binintersectSubset1) (forall (A:fofType) (B:fofType), ((((eq fofType) ((binintersect A) B)) A)->((subset A) B))))
% Defined: binintersectSubset1:=(forall (A:fofType) (B:fofType), ((((eq fofType) ((binintersect A) B)) A)->((subset A) B)))
% FOF formula (<kernel.Constant object at 0xbc95f0>, <kernel.Sort object at 0xc80f38>) of role type named bs114d_type
% Using role type
% Declaring bs114d:Prop
% FOF formula (((eq Prop) bs114d) (forall (A:fofType) (B:fofType) (C:fofType), (((eq fofType) ((binintersect A) ((binunion B) C))) ((binunion ((binintersect A) B)) ((binintersect A) C))))) of role definition named bs114d
% A new definition: (((eq Prop) bs114d) (forall (A:fofType) (B:fofType) (C:fofType), (((eq fofType) ((binintersect A) ((binunion B) C))) ((binunion ((binintersect A) B)) ((binintersect A) C)))))
% Defined: bs114d:=(forall (A:fofType) (B:fofType) (C:fofType), (((eq fofType) ((binintersect A) ((binunion B) C))) ((binunion ((binintersect A) B)) ((binintersect A) C))))
% FOF formula (<kernel.Constant object at 0xbc9710>, <kernel.DependentProduct object at 0xbc97a0>) of role type named regular_type
% Using role type
% Declaring regular:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xbc9128>, <kernel.DependentProduct object at 0xbc92d8>) of role type named setminus_type
% Using role type
% Declaring setminus:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0xbc95f0>, <kernel.Sort object at 0xc80f38>) of role type named setminusI_type
% Using role type
% Declaring setminusI:Prop
% FOF formula (((eq Prop) setminusI) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->((in Xx) ((setminus A) B)))))) of role definition named setminusI
% A new definition: (((eq Prop) setminusI) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->((in Xx) ((setminus A) B))))))
% Defined: setminusI:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->((in Xx) ((setminus A) B)))))
% FOF formula (<kernel.Constant object at 0xbc9680>, <kernel.Sort object at 0xc80f38>) of role type named setminusEL_type
% Using role type
% Declaring setminusEL:Prop
% FOF formula (((eq Prop) setminusEL) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((setminus A) B))->((in Xx) A)))) of role definition named setminusEL
% A new definition: (((eq Prop) setminusEL) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((setminus A) B))->((in Xx) A))))
% Defined: setminusEL:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((setminus A) B))->((in Xx) A)))
% FOF formula (<kernel.Constant object at 0xbc9878>, <kernel.Sort object at 0xc80f38>) of role type named setminusER_type
% Using role type
% Declaring setminusER:Prop
% FOF formula (((eq Prop) setminusER) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((setminus A) B))->(((in Xx) B)->False)))) of role definition named setminusER
% A new definition: (((eq Prop) setminusER) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((setminus A) B))->(((in Xx) B)->False))))
% Defined: setminusER:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((setminus A) B))->(((in Xx) B)->False)))
% FOF formula (<kernel.Constant object at 0xbc9a28>, <kernel.Sort object at 0xc80f38>) of role type named setminusSubset2_type
% Using role type
% Declaring setminusSubset2:Prop
% FOF formula (((eq Prop) setminusSubset2) (forall (A:fofType) (B:fofType), (((subset A) B)->(((eq fofType) ((setminus A) B)) emptyset)))) of role definition named setminusSubset2
% A new definition: (((eq Prop) setminusSubset2) (forall (A:fofType) (B:fofType), (((subset A) B)->(((eq fofType) ((setminus A) B)) emptyset))))
% Defined: setminusSubset2:=(forall (A:fofType) (B:fofType), (((subset A) B)->(((eq fofType) ((setminus A) B)) emptyset)))
% FOF formula (<kernel.Constant object at 0xbc9950>, <kernel.Sort object at 0xc80f38>) of role type named setminusERneg_type
% Using role type
% Declaring setminusERneg:Prop
% FOF formula (((eq Prop) setminusERneg) (forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) ((setminus A) B))->False)->(((in Xx) A)->((in Xx) B))))) of role definition named setminusERneg
% A new definition: (((eq Prop) setminusERneg) (forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) ((setminus A) B))->False)->(((in Xx) A)->((in Xx) B)))))
% Defined: setminusERneg:=(forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) ((setminus A) B))->False)->(((in Xx) A)->((in Xx) B))))
% FOF formula (<kernel.Constant object at 0xbc9c20>, <kernel.Sort object at 0xc80f38>) of role type named setminusELneg_type
% Using role type
% Declaring setminusELneg:Prop
% FOF formula (((eq Prop) setminusELneg) (forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) ((setminus A) B))->False)->((((in Xx) B)->False)->(((in Xx) A)->False))))) of role definition named setminusELneg
% A new definition: (((eq Prop) setminusELneg) (forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) ((setminus A) B))->False)->((((in Xx) B)->False)->(((in Xx) A)->False)))))
% Defined: setminusELneg:=(forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) ((setminus A) B))->False)->((((in Xx) B)->False)->(((in Xx) A)->False))))
% FOF formula (<kernel.Constant object at 0xbc9248>, <kernel.Sort object at 0xc80f38>) of role type named setminusILneg_type
% Using role type
% Declaring setminusILneg:Prop
% FOF formula (((eq Prop) setminusILneg) (forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) A)->False)->(((in Xx) ((setminus A) B))->False)))) of role definition named setminusILneg
% A new definition: (((eq Prop) setminusILneg) (forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) A)->False)->(((in Xx) ((setminus A) B))->False))))
% Defined: setminusILneg:=(forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) A)->False)->(((in Xx) ((setminus A) B))->False)))
% FOF formula (<kernel.Constant object at 0xbc9098>, <kernel.Sort object at 0xc80f38>) of role type named setminusIRneg_type
% Using role type
% Declaring setminusIRneg:Prop
% FOF formula (((eq Prop) setminusIRneg) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) B)->(((in Xx) ((setminus A) B))->False)))) of role definition named setminusIRneg
% A new definition: (((eq Prop) setminusIRneg) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) B)->(((in Xx) ((setminus A) B))->False))))
% Defined: setminusIRneg:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) B)->(((in Xx) ((setminus A) B))->False)))
% FOF formula (<kernel.Constant object at 0xbc9b48>, <kernel.Sort object at 0xc80f38>) of role type named setminusLsub_type
% Using role type
% Declaring setminusLsub:Prop
% FOF formula (((eq Prop) setminusLsub) (forall (A:fofType) (B:fofType), ((subset ((setminus A) B)) A))) of role definition named setminusLsub
% A new definition: (((eq Prop) setminusLsub) (forall (A:fofType) (B:fofType), ((subset ((setminus A) B)) A)))
% Defined: setminusLsub:=(forall (A:fofType) (B:fofType), ((subset ((setminus A) B)) A))
% FOF formula (<kernel.Constant object at 0xbc9638>, <kernel.Sort object at 0xc80f38>) of role type named setminusSubset1_type
% Using role type
% Declaring setminusSubset1:Prop
% FOF formula (((eq Prop) setminusSubset1) (forall (A:fofType) (B:fofType), ((((eq fofType) ((setminus A) B)) emptyset)->((subset A) B)))) of role definition named setminusSubset1
% A new definition: (((eq Prop) setminusSubset1) (forall (A:fofType) (B:fofType), ((((eq fofType) ((setminus A) B)) emptyset)->((subset A) B))))
% Defined: setminusSubset1:=(forall (A:fofType) (B:fofType), ((((eq fofType) ((setminus A) B)) emptyset)->((subset A) B)))
% FOF formula (<kernel.Constant object at 0xbc9d88>, <kernel.DependentProduct object at 0xbc9ea8>) of role type named symdiff_type
% Using role type
% Declaring symdiff:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0xbc9fc8>, <kernel.Sort object at 0xc80f38>) of role type named symdiffE_type
% Using role type
% Declaring symdiffE:Prop
% FOF formula (((eq Prop) symdiffE) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((symdiff A) B))->(forall (Xphi:Prop), ((((in Xx) A)->((((in Xx) B)->False)->Xphi))->(((((in Xx) A)->False)->(((in Xx) B)->Xphi))->Xphi)))))) of role definition named symdiffE
% A new definition: (((eq Prop) symdiffE) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((symdiff A) B))->(forall (Xphi:Prop), ((((in Xx) A)->((((in Xx) B)->False)->Xphi))->(((((in Xx) A)->False)->(((in Xx) B)->Xphi))->Xphi))))))
% Defined: symdiffE:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((symdiff A) B))->(forall (Xphi:Prop), ((((in Xx) A)->((((in Xx) B)->False)->Xphi))->(((((in Xx) A)->False)->(((in Xx) B)->Xphi))->Xphi)))))
% FOF formula (<kernel.Constant object at 0xbc9638>, <kernel.Sort object at 0xc80f38>) of role type named symdiffI1_type
% Using role type
% Declaring symdiffI1:Prop
% FOF formula (((eq Prop) symdiffI1) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->((in Xx) ((symdiff A) B)))))) of role definition named symdiffI1
% A new definition: (((eq Prop) symdiffI1) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->((in Xx) ((symdiff A) B))))))
% Defined: symdiffI1:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->((in Xx) ((symdiff A) B)))))
% FOF formula (<kernel.Constant object at 0xbc97a0>, <kernel.Sort object at 0xc80f38>) of role type named symdiffI2_type
% Using role type
% Declaring symdiffI2:Prop
% FOF formula (((eq Prop) symdiffI2) (forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) A)->False)->(((in Xx) B)->((in Xx) ((symdiff A) B)))))) of role definition named symdiffI2
% A new definition: (((eq Prop) symdiffI2) (forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) A)->False)->(((in Xx) B)->((in Xx) ((symdiff A) B))))))
% Defined: symdiffI2:=(forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) A)->False)->(((in Xx) B)->((in Xx) ((symdiff A) B)))))
% FOF formula (<kernel.Constant object at 0xbc9908>, <kernel.Sort object at 0xc80f38>) of role type named symdiffIneg1_type
% Using role type
% Declaring symdiffIneg1:Prop
% FOF formula (((eq Prop) symdiffIneg1) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(((in Xx) B)->(((in Xx) ((symdiff A) B))->False))))) of role definition named symdiffIneg1
% A new definition: (((eq Prop) symdiffIneg1) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(((in Xx) B)->(((in Xx) ((symdiff A) B))->False)))))
% Defined: symdiffIneg1:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(((in Xx) B)->(((in Xx) ((symdiff A) B))->False))))
% FOF formula (<kernel.Constant object at 0xbc9710>, <kernel.Sort object at 0xc80f38>) of role type named symdiffIneg2_type
% Using role type
% Declaring symdiffIneg2:Prop
% FOF formula (((eq Prop) symdiffIneg2) (forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) A)->False)->((((in Xx) B)->False)->(((in Xx) ((symdiff A) B))->False))))) of role definition named symdiffIneg2
% A new definition: (((eq Prop) symdiffIneg2) (forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) A)->False)->((((in Xx) B)->False)->(((in Xx) ((symdiff A) B))->False)))))
% Defined: symdiffIneg2:=(forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) A)->False)->((((in Xx) B)->False)->(((in Xx) ((symdiff A) B))->False))))
% FOF formula (<kernel.Constant object at 0xbc9f80>, <kernel.DependentProduct object at 0xbc9c68>) of role type named iskpair_type
% Using role type
% Declaring iskpair:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xbc9b90>, <kernel.Sort object at 0xc80f38>) of role type named secondinupair_type
% Using role type
% Declaring secondinupair:Prop
% FOF formula (((eq Prop) secondinupair) (forall (Xx:fofType) (Xy:fofType), ((in Xy) ((setadjoin Xx) ((setadjoin Xy) emptyset))))) of role definition named secondinupair
% A new definition: (((eq Prop) secondinupair) (forall (Xx:fofType) (Xy:fofType), ((in Xy) ((setadjoin Xx) ((setadjoin Xy) emptyset)))))
% Defined: secondinupair:=(forall (Xx:fofType) (Xy:fofType), ((in Xy) ((setadjoin Xx) ((setadjoin Xy) emptyset))))
% FOF formula (<kernel.Constant object at 0xbc9710>, <kernel.Sort object at 0xc80f38>) of role type named setukpairIL_type
% Using role type
% Declaring setukpairIL:Prop
% FOF formula (((eq Prop) setukpairIL) (forall (Xx:fofType) (Xy:fofType), ((in Xx) (setunion ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))))) of role definition named setukpairIL
% A new definition: (((eq Prop) setukpairIL) (forall (Xx:fofType) (Xy:fofType), ((in Xx) (setunion ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))))))
% Defined: setukpairIL:=(forall (Xx:fofType) (Xy:fofType), ((in Xx) (setunion ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))))
% FOF formula (<kernel.Constant object at 0xbc94d0>, <kernel.Sort object at 0xc80f38>) of role type named setukpairIR_type
% Using role type
% Declaring setukpairIR:Prop
% FOF formula (((eq Prop) setukpairIR) (forall (Xx:fofType) (Xy:fofType), ((in Xy) (setunion ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))))) of role definition named setukpairIR
% A new definition: (((eq Prop) setukpairIR) (forall (Xx:fofType) (Xy:fofType), ((in Xy) (setunion ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))))))
% Defined: setukpairIR:=(forall (Xx:fofType) (Xy:fofType), ((in Xy) (setunion ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))))
% FOF formula (<kernel.Constant object at 0xbc90e0>, <kernel.Sort object at 0xc80f38>) of role type named kpairiskpair_type
% Using role type
% Declaring kpairiskpair:Prop
% FOF formula (((eq Prop) kpairiskpair) (forall (Xx:fofType) (Xy:fofType), (iskpair ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))))) of role definition named kpairiskpair
% A new definition: (((eq Prop) kpairiskpair) (forall (Xx:fofType) (Xy:fofType), (iskpair ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))))
% Defined: kpairiskpair:=(forall (Xx:fofType) (Xy:fofType), (iskpair ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))))
% FOF formula (<kernel.Constant object at 0xbc90e0>, <kernel.DependentProduct object at 0x9f6290>) of role type named kpair_type
% Using role type
% Declaring kpair:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0xbc94d0>, <kernel.Sort object at 0xc80f38>) of role type named kpairp_type
% Using role type
% Declaring kpairp:Prop
% FOF formula (((eq Prop) kpairp) (forall (Xx:fofType) (Xy:fofType), (iskpair ((kpair Xx) Xy)))) of role definition named kpairp
% A new definition: (((eq Prop) kpairp) (forall (Xx:fofType) (Xy:fofType), (iskpair ((kpair Xx) Xy))))
% Defined: kpairp:=(forall (Xx:fofType) (Xy:fofType), (iskpair ((kpair Xx) Xy)))
% FOF formula (<kernel.Constant object at 0xbc90e0>, <kernel.DependentProduct object at 0x9f60e0>) of role type named cartprod_type
% Using role type
% Declaring cartprod:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0x9f6320>, <kernel.Sort object at 0xc80f38>) of role type named singletonsubset_type
% Using role type
% Declaring singletonsubset:Prop
% FOF formula (((eq Prop) singletonsubset) (forall (A:fofType) (Xx:fofType), (((in Xx) A)->((subset ((setadjoin Xx) emptyset)) A)))) of role definition named singletonsubset
% A new definition: (((eq Prop) singletonsubset) (forall (A:fofType) (Xx:fofType), (((in Xx) A)->((subset ((setadjoin Xx) emptyset)) A))))
% Defined: singletonsubset:=(forall (A:fofType) (Xx:fofType), (((in Xx) A)->((subset ((setadjoin Xx) emptyset)) A)))
% FOF formula (<kernel.Constant object at 0x9f6290>, <kernel.Sort object at 0xc80f38>) of role type named singletoninpowerset_type
% Using role type
% Declaring singletoninpowerset:Prop
% FOF formula (((eq Prop) singletoninpowerset) (forall (A:fofType) (Xx:fofType), (((in Xx) A)->((in ((setadjoin Xx) emptyset)) (powerset A))))) of role definition named singletoninpowerset
% A new definition: (((eq Prop) singletoninpowerset) (forall (A:fofType) (Xx:fofType), (((in Xx) A)->((in ((setadjoin Xx) emptyset)) (powerset A)))))
% Defined: singletoninpowerset:=(forall (A:fofType) (Xx:fofType), (((in Xx) A)->((in ((setadjoin Xx) emptyset)) (powerset A))))
% FOF formula (<kernel.Constant object at 0x9f6098>, <kernel.Sort object at 0xc80f38>) of role type named singletoninpowunion_type
% Using role type
% Declaring singletoninpowunion:Prop
% FOF formula (((eq Prop) singletoninpowunion) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((in ((setadjoin Xx) emptyset)) (powerset ((binunion A) B)))))) of role definition named singletoninpowunion
% A new definition: (((eq Prop) singletoninpowunion) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((in ((setadjoin Xx) emptyset)) (powerset ((binunion A) B))))))
% Defined: singletoninpowunion:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((in ((setadjoin Xx) emptyset)) (powerset ((binunion A) B)))))
% FOF formula (<kernel.Constant object at 0x9f6200>, <kernel.Sort object at 0xc80f38>) of role type named upairset2E_type
% Using role type
% Declaring upairset2E:Prop
% FOF formula (((eq Prop) upairset2E) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((in Xz) ((setadjoin Xx) ((setadjoin Xy) emptyset)))->((or (((eq fofType) Xz) Xx)) (((eq fofType) Xz) Xy))))) of role definition named upairset2E
% A new definition: (((eq Prop) upairset2E) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((in Xz) ((setadjoin Xx) ((setadjoin Xy) emptyset)))->((or (((eq fofType) Xz) Xx)) (((eq fofType) Xz) Xy)))))
% Defined: upairset2E:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((in Xz) ((setadjoin Xx) ((setadjoin Xy) emptyset)))->((or (((eq fofType) Xz) Xx)) (((eq fofType) Xz) Xy))))
% FOF formula (<kernel.Constant object at 0x9f6248>, <kernel.Sort object at 0xc80f38>) of role type named upairsubunion_type
% Using role type
% Declaring upairsubunion:Prop
% FOF formula (((eq Prop) upairsubunion) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((subset ((setadjoin Xx) ((setadjoin Xy) emptyset))) ((binunion A) B))))))) of role definition named upairsubunion
% A new definition: (((eq Prop) upairsubunion) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((subset ((setadjoin Xx) ((setadjoin Xy) emptyset))) ((binunion A) B)))))))
% Defined: upairsubunion:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((subset ((setadjoin Xx) ((setadjoin Xy) emptyset))) ((binunion A) B))))))
% FOF formula (<kernel.Constant object at 0x9f6710>, <kernel.Sort object at 0xc80f38>) of role type named upairinpowunion_type
% Using role type
% Declaring upairinpowunion:Prop
% FOF formula (((eq Prop) upairinpowunion) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((setadjoin Xx) ((setadjoin Xy) emptyset))) (powerset ((binunion A) B)))))))) of role definition named upairinpowunion
% A new definition: (((eq Prop) upairinpowunion) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((setadjoin Xx) ((setadjoin Xy) emptyset))) (powerset ((binunion A) B))))))))
% Defined: upairinpowunion:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((setadjoin Xx) ((setadjoin Xy) emptyset))) (powerset ((binunion A) B)))))))
% FOF formula (<kernel.Constant object at 0x9f6638>, <kernel.Sort object at 0xc80f38>) of role type named ubforcartprodlem1_type
% Using role type
% Declaring ubforcartprodlem1:Prop
% FOF formula (((eq Prop) ubforcartprodlem1) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((subset ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) (powerset ((binunion A) B)))))))) of role definition named ubforcartprodlem1
% A new definition: (((eq Prop) ubforcartprodlem1) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((subset ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) (powerset ((binunion A) B))))))))
% Defined: ubforcartprodlem1:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((subset ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) (powerset ((binunion A) B)))))))
% FOF formula (<kernel.Constant object at 0x9f67a0>, <kernel.Sort object at 0xc80f38>) of role type named ubforcartprodlem2_type
% Using role type
% Declaring ubforcartprodlem2:Prop
% FOF formula (((eq Prop) ubforcartprodlem2) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) (powerset (powerset ((binunion A) B))))))))) of role definition named ubforcartprodlem2
% A new definition: (((eq Prop) ubforcartprodlem2) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) (powerset (powerset ((binunion A) B)))))))))
% Defined: ubforcartprodlem2:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) (powerset (powerset ((binunion A) B))))))))
% FOF formula (<kernel.Constant object at 0x9f6488>, <kernel.Sort object at 0xc80f38>) of role type named ubforcartprodlem3_type
% Using role type
% Declaring ubforcartprodlem3:Prop
% FOF formula (((eq Prop) ubforcartprodlem3) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))))))) of role definition named ubforcartprodlem3
% A new definition: (((eq Prop) ubforcartprodlem3) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))))))))
% Defined: ubforcartprodlem3:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))))))
% FOF formula (<kernel.Constant object at 0x9f6050>, <kernel.Sort object at 0xc80f38>) of role type named cartprodpairin_type
% Using role type
% Declaring cartprodpairin:Prop
% FOF formula (((eq Prop) cartprodpairin) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))))))) of role definition named cartprodpairin
% A new definition: (((eq Prop) cartprodpairin) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))))
% Defined: cartprodpairin:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))))))
% FOF formula (<kernel.Constant object at 0x9f6560>, <kernel.Sort object at 0xc80f38>) of role type named cartprodmempair1_type
% Using role type
% Declaring cartprodmempair1:Prop
% FOF formula (((eq Prop) cartprodmempair1) (forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) B)) (((eq fofType) Xu) ((kpair Xx) Xy))))))))))) of role definition named cartprodmempair1
% A new definition: (((eq Prop) cartprodmempair1) (forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) B)) (((eq fofType) Xu) ((kpair Xx) Xy)))))))))))
% Defined: cartprodmempair1:=(forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) B)) (((eq fofType) Xu) ((kpair Xx) Xy))))))))))
% FOF formula (<kernel.Constant object at 0x9f6908>, <kernel.Sort object at 0xc80f38>) of role type named cartprodmempair_type
% Using role type
% Declaring cartprodmempair:Prop
% FOF formula (((eq Prop) cartprodmempair) (forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->(iskpair Xu)))) of role definition named cartprodmempair
% A new definition: (((eq Prop) cartprodmempair) (forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->(iskpair Xu))))
% Defined: cartprodmempair:=(forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->(iskpair Xu)))
% FOF formula (<kernel.Constant object at 0x9f69e0>, <kernel.Sort object at 0xc80f38>) of role type named setunionE2_type
% Using role type
% Declaring setunionE2:Prop
% FOF formula (((eq Prop) setunionE2) (forall (A:fofType) (Xx:fofType), (((in Xx) (setunion A))->((ex fofType) (fun (X:fofType)=> ((and ((in X) A)) ((in Xx) X))))))) of role definition named setunionE2
% A new definition: (((eq Prop) setunionE2) (forall (A:fofType) (Xx:fofType), (((in Xx) (setunion A))->((ex fofType) (fun (X:fofType)=> ((and ((in X) A)) ((in Xx) X)))))))
% Defined: setunionE2:=(forall (A:fofType) (Xx:fofType), (((in Xx) (setunion A))->((ex fofType) (fun (X:fofType)=> ((and ((in X) A)) ((in Xx) X))))))
% FOF formula (<kernel.Constant object at 0x9f6518>, <kernel.Sort object at 0xc80f38>) of role type named setunionsingleton1_type
% Using role type
% Declaring setunionsingleton1:Prop
% FOF formula (((eq Prop) setunionsingleton1) (forall (A:fofType), ((subset (setunion ((setadjoin A) emptyset))) A))) of role definition named setunionsingleton1
% A new definition: (((eq Prop) setunionsingleton1) (forall (A:fofType), ((subset (setunion ((setadjoin A) emptyset))) A)))
% Defined: setunionsingleton1:=(forall (A:fofType), ((subset (setunion ((setadjoin A) emptyset))) A))
% FOF formula (<kernel.Constant object at 0x9f6a28>, <kernel.Sort object at 0xc80f38>) of role type named setunionsingleton2_type
% Using role type
% Declaring setunionsingleton2:Prop
% FOF formula (((eq Prop) setunionsingleton2) (forall (A:fofType), ((subset A) (setunion ((setadjoin A) emptyset))))) of role definition named setunionsingleton2
% A new definition: (((eq Prop) setunionsingleton2) (forall (A:fofType), ((subset A) (setunion ((setadjoin A) emptyset)))))
% Defined: setunionsingleton2:=(forall (A:fofType), ((subset A) (setunion ((setadjoin A) emptyset))))
% FOF formula (<kernel.Constant object at 0x9f6f80>, <kernel.Sort object at 0xc80f38>) of role type named setunionsingleton_type
% Using role type
% Declaring setunionsingleton:Prop
% FOF formula (((eq Prop) setunionsingleton) (forall (Xx:fofType), (((eq fofType) (setunion ((setadjoin Xx) emptyset))) Xx))) of role definition named setunionsingleton
% A new definition: (((eq Prop) setunionsingleton) (forall (Xx:fofType), (((eq fofType) (setunion ((setadjoin Xx) emptyset))) Xx)))
% Defined: setunionsingleton:=(forall (Xx:fofType), (((eq fofType) (setunion ((setadjoin Xx) emptyset))) Xx))
% FOF formula (<kernel.Constant object at 0x9f6dd0>, <kernel.DependentProduct object at 0x9f6758>) of role type named singleton_type
% Using role type
% Declaring singleton:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x9f6b90>, <kernel.Sort object at 0xc80f38>) of role type named singletonprop_type
% Using role type
% Declaring singletonprop:Prop
% FOF formula (((eq Prop) singletonprop) (forall (A:fofType) (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))))->(((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->(singleton ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))))))) of role definition named singletonprop
% A new definition: (((eq Prop) singletonprop) (forall (A:fofType) (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))))->(((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->(singleton ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx))))))))
% Defined: singletonprop:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))))->(((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->(singleton ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))))))
% FOF formula (<kernel.Constant object at 0x9f6f80>, <kernel.DependentProduct object at 0x9f6ef0>) of role type named ex1_type
% Using role type
% Declaring ex1:(fofType->((fofType->Prop)->Prop))
% FOF formula (<kernel.Constant object at 0x9f6fc8>, <kernel.Sort object at 0xc80f38>) of role type named ex1E1_type
% Using role type
% Declaring ex1E1:Prop
% FOF formula (((eq Prop) ex1E1) (forall (A:fofType) (Xphi:(fofType->Prop)), (((ex1 A) (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))))) of role definition named ex1E1
% A new definition: (((eq Prop) ex1E1) (forall (A:fofType) (Xphi:(fofType->Prop)), (((ex1 A) (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx)))))))
% Defined: ex1E1:=(forall (A:fofType) (Xphi:(fofType->Prop)), (((ex1 A) (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))))
% FOF formula (<kernel.Constant object at 0x9f6b90>, <kernel.Sort object at 0xc80f38>) of role type named ex1I_type
% Using role type
% Declaring ex1I:Prop
% FOF formula (((eq Prop) ex1I) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((forall (Xy:fofType), (((in Xy) A)->((Xphi Xy)->(((eq fofType) Xy) Xx))))->((ex1 A) (fun (Xy:fofType)=> (Xphi Xy)))))))) of role definition named ex1I
% A new definition: (((eq Prop) ex1I) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((forall (Xy:fofType), (((in Xy) A)->((Xphi Xy)->(((eq fofType) Xy) Xx))))->((ex1 A) (fun (Xy:fofType)=> (Xphi Xy))))))))
% Defined: ex1I:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((forall (Xy:fofType), (((in Xy) A)->((Xphi Xy)->(((eq fofType) Xy) Xx))))->((ex1 A) (fun (Xy:fofType)=> (Xphi Xy)))))))
% FOF formula (<kernel.Constant object at 0x9f6c68>, <kernel.Sort object at 0xc80f38>) of role type named ex1I2_type
% Using role type
% Declaring ex1I2:Prop
% FOF formula (((eq Prop) ex1I2) (forall (A:fofType) (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))))->(((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->((ex1 A) (fun (Xx:fofType)=> (Xphi Xx))))))) of role definition named ex1I2
% A new definition: (((eq Prop) ex1I2) (forall (A:fofType) (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))))->(((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->((ex1 A) (fun (Xx:fofType)=> (Xphi Xx)))))))
% Defined: ex1I2:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))))->(((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->((ex1 A) (fun (Xx:fofType)=> (Xphi Xx))))))
% FOF formula (<kernel.Constant object at 0x9f6170>, <kernel.Sort object at 0xc80f38>) of role type named singletonsuniq_type
% Using role type
% Declaring singletonsuniq:Prop
% FOF formula (((eq Prop) singletonsuniq) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) ((setadjoin Xx) emptyset)) ((setadjoin Xy) emptyset))->(((eq fofType) Xx) Xy)))) of role definition named singletonsuniq
% A new definition: (((eq Prop) singletonsuniq) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) ((setadjoin Xx) emptyset)) ((setadjoin Xy) emptyset))->(((eq fofType) Xx) Xy))))
% Defined: singletonsuniq:=(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) ((setadjoin Xx) emptyset)) ((setadjoin Xy) emptyset))->(((eq fofType) Xx) Xy)))
% FOF formula (<kernel.Constant object at 0x9f6b48>, <kernel.DependentProduct object at 0x9f6f80>) of role type named atmost1p_type
% Using role type
% Declaring atmost1p:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x9f62d8>, <kernel.DependentProduct object at 0x9f6a70>) of role type named atleast2p_type
% Using role type
% Declaring atleast2p:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x9f6170>, <kernel.DependentProduct object at 0x9f6e18>) of role type named atmost2p_type
% Using role type
% Declaring atmost2p:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x9f6878>, <kernel.DependentProduct object at 0x9f7440>) of role type named upairsetp_type
% Using role type
% Declaring upairsetp:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x9f6f80>, <kernel.Sort object at 0xc80f38>) of role type named setukpairinjL1_type
% Using role type
% Declaring setukpairinjL1:Prop
% FOF formula (((eq Prop) setukpairinjL1) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((in ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))->(((eq fofType) Xx) Xz)))) of role definition named setukpairinjL1
% A new definition: (((eq Prop) setukpairinjL1) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((in ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))->(((eq fofType) Xx) Xz))))
% Defined: setukpairinjL1:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((in ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))->(((eq fofType) Xx) Xz)))
% FOF formula (<kernel.Constant object at 0x9f6d88>, <kernel.Sort object at 0xc80f38>) of role type named kfstsingleton_type
% Using role type
% Declaring kfstsingleton:Prop
% FOF formula (((eq Prop) kfstsingleton) (forall (Xu:fofType), ((iskpair Xu)->(singleton ((dsetconstr (setunion Xu)) (fun (Xx:fofType)=> ((in ((setadjoin Xx) emptyset)) Xu))))))) of role definition named kfstsingleton
% A new definition: (((eq Prop) kfstsingleton) (forall (Xu:fofType), ((iskpair Xu)->(singleton ((dsetconstr (setunion Xu)) (fun (Xx:fofType)=> ((in ((setadjoin Xx) emptyset)) Xu)))))))
% Defined: kfstsingleton:=(forall (Xu:fofType), ((iskpair Xu)->(singleton ((dsetconstr (setunion Xu)) (fun (Xx:fofType)=> ((in ((setadjoin Xx) emptyset)) Xu))))))
% FOF formula (<kernel.Constant object at 0x9f6d88>, <kernel.Sort object at 0xc80f38>) of role type named theprop_type
% Using role type
% Declaring theprop:Prop
% FOF formula (((eq Prop) theprop) (forall (X:fofType), ((singleton X)->((in (setunion X)) X)))) of role definition named theprop
% A new definition: (((eq Prop) theprop) (forall (X:fofType), ((singleton X)->((in (setunion X)) X))))
% Defined: theprop:=(forall (X:fofType), ((singleton X)->((in (setunion X)) X)))
% FOF formula (<kernel.Constant object at 0x9f6d88>, <kernel.DependentProduct object at 0x9f7200>) of role type named kfst_type
% Using role type
% Declaring kfst:(fofType->fofType)
% FOF formula (<kernel.Constant object at 0x9f6d88>, <kernel.Sort object at 0xc80f38>) of role type named kfstpairEq_type
% Using role type
% Declaring kfstpairEq:Prop
% FOF formula (((eq Prop) kfstpairEq) (forall (Xx:fofType) (Xy:fofType), (((eq fofType) (kfst ((kpair Xx) Xy))) Xx))) of role definition named kfstpairEq
% A new definition: (((eq Prop) kfstpairEq) (forall (Xx:fofType) (Xy:fofType), (((eq fofType) (kfst ((kpair Xx) Xy))) Xx)))
% Defined: kfstpairEq:=(forall (Xx:fofType) (Xy:fofType), (((eq fofType) (kfst ((kpair Xx) Xy))) Xx))
% FOF formula (<kernel.Constant object at 0x9f7830>, <kernel.Sort object at 0xc80f38>) of role type named cartprodfstin_type
% Using role type
% Declaring cartprodfstin:Prop
% FOF formula (((eq Prop) cartprodfstin) (forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->((in (kfst Xu)) A)))) of role definition named cartprodfstin
% A new definition: (((eq Prop) cartprodfstin) (forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->((in (kfst Xu)) A))))
% Defined: cartprodfstin:=(forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->((in (kfst Xu)) A)))
% FOF formula (<kernel.Constant object at 0x9f7290>, <kernel.Sort object at 0xc80f38>) of role type named setukpairinjL2_type
% Using role type
% Declaring setukpairinjL2:Prop
% FOF formula (((eq Prop) setukpairinjL2) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xz) ((setadjoin Xu) emptyset))) emptyset)))->(((eq fofType) Xx) Xz)))) of role definition named setukpairinjL2
% A new definition: (((eq Prop) setukpairinjL2) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xz) ((setadjoin Xu) emptyset))) emptyset)))->(((eq fofType) Xx) Xz))))
% Defined: setukpairinjL2:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xz) ((setadjoin Xu) emptyset))) emptyset)))->(((eq fofType) Xx) Xz)))
% FOF formula (<kernel.Constant object at 0x9f7758>, <kernel.Sort object at 0xc80f38>) of role type named setukpairinjL_type
% Using role type
% Declaring setukpairinjL:Prop
% FOF formula (((eq Prop) setukpairinjL) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((kpair Xx) Xy)) ((kpair Xz) Xu))->(((eq fofType) Xx) Xz)))) of role definition named setukpairinjL
% A new definition: (((eq Prop) setukpairinjL) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((kpair Xx) Xy)) ((kpair Xz) Xu))->(((eq fofType) Xx) Xz))))
% Defined: setukpairinjL:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((kpair Xx) Xy)) ((kpair Xz) Xu))->(((eq fofType) Xx) Xz)))
% FOF formula (<kernel.Constant object at 0x9f7098>, <kernel.Sort object at 0xc80f38>) of role type named setukpairinjR11_type
% Using role type
% Declaring setukpairinjR11:Prop
% FOF formula (((eq Prop) setukpairinjR11) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(((eq fofType) ((setadjoin Xx) ((setadjoin Xy) emptyset))) ((setadjoin Xx) emptyset))))) of role definition named setukpairinjR11
% A new definition: (((eq Prop) setukpairinjR11) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(((eq fofType) ((setadjoin Xx) ((setadjoin Xy) emptyset))) ((setadjoin Xx) emptyset)))))
% Defined: setukpairinjR11:=(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(((eq fofType) ((setadjoin Xx) ((setadjoin Xy) emptyset))) ((setadjoin Xx) emptyset))))
% FOF formula (<kernel.Constant object at 0x9f74d0>, <kernel.Sort object at 0xc80f38>) of role type named setukpairinjR12_type
% Using role type
% Declaring setukpairinjR12:Prop
% FOF formula (((eq Prop) setukpairinjR12) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xx) emptyset)) emptyset))))) of role definition named setukpairinjR12
% A new definition: (((eq Prop) setukpairinjR12) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xx) emptyset)) emptyset)))))
% Defined: setukpairinjR12:=(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xx) emptyset)) emptyset))))
% FOF formula (<kernel.Constant object at 0x9f7128>, <kernel.Sort object at 0xc80f38>) of role type named setukpairinjR1_type
% Using role type
% Declaring setukpairinjR1:Prop
% FOF formula (((eq Prop) setukpairinjR1) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xz) ((setadjoin Xu) emptyset))) emptyset)))->((((eq fofType) Xz) Xu)->(((eq fofType) Xy) Xu))))) of role definition named setukpairinjR1
% A new definition: (((eq Prop) setukpairinjR1) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xz) ((setadjoin Xu) emptyset))) emptyset)))->((((eq fofType) Xz) Xu)->(((eq fofType) Xy) Xu)))))
% Defined: setukpairinjR1:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xz) ((setadjoin Xu) emptyset))) emptyset)))->((((eq fofType) Xz) Xu)->(((eq fofType) Xy) Xu))))
% FOF formula (<kernel.Constant object at 0x9f7638>, <kernel.Sort object at 0xc80f38>) of role type named upairequniteq_type
% Using role type
% Declaring upairequniteq:Prop
% FOF formula (((eq Prop) upairequniteq) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), ((((eq fofType) ((setadjoin Xx) ((setadjoin Xy) emptyset))) ((setadjoin Xz) emptyset))->(((eq fofType) Xx) Xy)))) of role definition named upairequniteq
% A new definition: (((eq Prop) upairequniteq) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), ((((eq fofType) ((setadjoin Xx) ((setadjoin Xy) emptyset))) ((setadjoin Xz) emptyset))->(((eq fofType) Xx) Xy))))
% Defined: upairequniteq:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), ((((eq fofType) ((setadjoin Xx) ((setadjoin Xy) emptyset))) ((setadjoin Xz) emptyset))->(((eq fofType) Xx) Xy)))
% FOF formula (<kernel.Constant object at 0x9f7998>, <kernel.Sort object at 0xc80f38>) of role type named setukpairinjR2_type
% Using role type
% Declaring setukpairinjR2:Prop
% FOF formula (((eq Prop) setukpairinjR2) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xz) ((setadjoin Xu) emptyset))) emptyset)))->(((eq fofType) Xy) Xu)))) of role definition named setukpairinjR2
% A new definition: (((eq Prop) setukpairinjR2) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xz) ((setadjoin Xu) emptyset))) emptyset)))->(((eq fofType) Xy) Xu))))
% Defined: setukpairinjR2:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xz) ((setadjoin Xu) emptyset))) emptyset)))->(((eq fofType) Xy) Xu)))
% FOF formula (<kernel.Constant object at 0x9f7440>, <kernel.Sort object at 0xc80f38>) of role type named setukpairinjR_type
% Using role type
% Declaring setukpairinjR:Prop
% FOF formula (((eq Prop) setukpairinjR) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((kpair Xx) Xy)) ((kpair Xz) Xu))->(((eq fofType) Xy) Xu)))) of role definition named setukpairinjR
% A new definition: (((eq Prop) setukpairinjR) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((kpair Xx) Xy)) ((kpair Xz) Xu))->(((eq fofType) Xy) Xu))))
% Defined: setukpairinjR:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((kpair Xx) Xy)) ((kpair Xz) Xu))->(((eq fofType) Xy) Xu)))
% FOF formula (<kernel.Constant object at 0x9f7908>, <kernel.Sort object at 0xc80f38>) of role type named ksndsingleton_type
% Using role type
% Declaring ksndsingleton:Prop
% FOF formula (((eq Prop) ksndsingleton) (forall (Xu:fofType), ((iskpair Xu)->(singleton ((dsetconstr (setunion Xu)) (fun (Xx:fofType)=> (((eq fofType) Xu) ((kpair (kfst Xu)) Xx)))))))) of role definition named ksndsingleton
% A new definition: (((eq Prop) ksndsingleton) (forall (Xu:fofType), ((iskpair Xu)->(singleton ((dsetconstr (setunion Xu)) (fun (Xx:fofType)=> (((eq fofType) Xu) ((kpair (kfst Xu)) Xx))))))))
% Defined: ksndsingleton:=(forall (Xu:fofType), ((iskpair Xu)->(singleton ((dsetconstr (setunion Xu)) (fun (Xx:fofType)=> (((eq fofType) Xu) ((kpair (kfst Xu)) Xx)))))))
% FOF formula (<kernel.Constant object at 0x9f7950>, <kernel.DependentProduct object at 0x9f7878>) of role type named ksnd_type
% Using role type
% Declaring ksnd:(fofType->fofType)
% FOF formula (<kernel.Constant object at 0x9f73f8>, <kernel.Sort object at 0xc80f38>) of role type named ksndpairEq_type
% Using role type
% Declaring ksndpairEq:Prop
% FOF formula (((eq Prop) ksndpairEq) (forall (Xx:fofType) (Xy:fofType), (((eq fofType) (ksnd ((kpair Xx) Xy))) Xy))) of role definition named ksndpairEq
% A new definition: (((eq Prop) ksndpairEq) (forall (Xx:fofType) (Xy:fofType), (((eq fofType) (ksnd ((kpair Xx) Xy))) Xy)))
% Defined: ksndpairEq:=(forall (Xx:fofType) (Xy:fofType), (((eq fofType) (ksnd ((kpair Xx) Xy))) Xy))
% FOF formula (<kernel.Constant object at 0x9f7908>, <kernel.Sort object at 0xc80f38>) of role type named kpairsurjEq_type
% Using role type
% Declaring kpairsurjEq:Prop
% FOF formula (((eq Prop) kpairsurjEq) (forall (Xu:fofType), ((iskpair Xu)->(((eq fofType) ((kpair (kfst Xu)) (ksnd Xu))) Xu)))) of role definition named kpairsurjEq
% A new definition: (((eq Prop) kpairsurjEq) (forall (Xu:fofType), ((iskpair Xu)->(((eq fofType) ((kpair (kfst Xu)) (ksnd Xu))) Xu))))
% Defined: kpairsurjEq:=(forall (Xu:fofType), ((iskpair Xu)->(((eq fofType) ((kpair (kfst Xu)) (ksnd Xu))) Xu)))
% FOF formula (<kernel.Constant object at 0x9f7950>, <kernel.Sort object at 0xc80f38>) of role type named cartprodsndin_type
% Using role type
% Declaring cartprodsndin:Prop
% FOF formula (((eq Prop) cartprodsndin) (forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->((in (ksnd Xu)) B)))) of role definition named cartprodsndin
% A new definition: (((eq Prop) cartprodsndin) (forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->((in (ksnd Xu)) B))))
% Defined: cartprodsndin:=(forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->((in (ksnd Xu)) B)))
% FOF formula (<kernel.Constant object at 0x9f7bd8>, <kernel.Sort object at 0xc80f38>) of role type named cartprodpairmemEL_type
% Using role type
% Declaring cartprodpairmemEL:Prop
% FOF formula (((eq Prop) cartprodpairmemEL) (forall (A:fofType) (B:fofType) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) ((cartprod A) B))->((in Xx) A)))) of role definition named cartprodpairmemEL
% A new definition: (((eq Prop) cartprodpairmemEL) (forall (A:fofType) (B:fofType) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) ((cartprod A) B))->((in Xx) A))))
% Defined: cartprodpairmemEL:=(forall (A:fofType) (B:fofType) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) ((cartprod A) B))->((in Xx) A)))
% FOF formula (<kernel.Constant object at 0x9f7560>, <kernel.Sort object at 0xc80f38>) of role type named cartprodpairmemER_type
% Using role type
% Declaring cartprodpairmemER:Prop
% FOF formula (((eq Prop) cartprodpairmemER) (forall (A:fofType) (B:fofType) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) ((cartprod A) B))->((in Xy) B)))) of role definition named cartprodpairmemER
% A new definition: (((eq Prop) cartprodpairmemER) (forall (A:fofType) (B:fofType) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) ((cartprod A) B))->((in Xy) B))))
% Defined: cartprodpairmemER:=(forall (A:fofType) (B:fofType) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) ((cartprod A) B))->((in Xy) B)))
% FOF formula (<kernel.Constant object at 0x9f7b48>, <kernel.Sort object at 0xc80f38>) of role type named cartprodmempaircEq_type
% Using role type
% Declaring cartprodmempaircEq:Prop
% FOF formula (((eq Prop) cartprodmempaircEq) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))))))) of role definition named cartprodmempaircEq
% A new definition: (((eq Prop) cartprodmempaircEq) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy)))))))
% Defined: cartprodmempaircEq:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))))))
% FOF formula (<kernel.Constant object at 0x9f7518>, <kernel.Sort object at 0xc80f38>) of role type named cartprodfstpairEq_type
% Using role type
% Declaring cartprodfstpairEq:Prop
% FOF formula (((eq Prop) cartprodfstpairEq) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((eq fofType) (kfst ((kpair Xx) Xy))) Xx)))))) of role definition named cartprodfstpairEq
% A new definition: (((eq Prop) cartprodfstpairEq) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((eq fofType) (kfst ((kpair Xx) Xy))) Xx))))))
% Defined: cartprodfstpairEq:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((eq fofType) (kfst ((kpair Xx) Xy))) Xx)))))
% FOF formula (<kernel.Constant object at 0x9f7488>, <kernel.Sort object at 0xc80f38>) of role type named cartprodsndpairEq_type
% Using role type
% Declaring cartprodsndpairEq:Prop
% FOF formula (((eq Prop) cartprodsndpairEq) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((eq fofType) (ksnd ((kpair Xx) Xy))) Xy)))))) of role definition named cartprodsndpairEq
% A new definition: (((eq Prop) cartprodsndpairEq) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((eq fofType) (ksnd ((kpair Xx) Xy))) Xy))))))
% Defined: cartprodsndpairEq:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((eq fofType) (ksnd ((kpair Xx) Xy))) Xy)))))
% FOF formula (<kernel.Constant object at 0x9f7d88>, <kernel.Sort object at 0xc80f38>) of role type named cartprodpairsurjEq_type
% Using role type
% Declaring cartprodpairsurjEq:Prop
% FOF formula (((eq Prop) cartprodpairsurjEq) (forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->(((eq fofType) ((kpair (kfst Xu)) (ksnd Xu))) Xu)))) of role definition named cartprodpairsurjEq
% A new definition: (((eq Prop) cartprodpairsurjEq) (forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->(((eq fofType) ((kpair (kfst Xu)) (ksnd Xu))) Xu))))
% Defined: cartprodpairsurjEq:=(forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->(((eq fofType) ((kpair (kfst Xu)) (ksnd Xu))) Xu)))
% FOF formula (<kernel.Constant object at 0x9f7e60>, <kernel.DependentProduct object at 0x9f7b90>) of role type named breln_type
% Using role type
% Declaring breln:(fofType->(fofType->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0x9f75a8>, <kernel.DependentProduct object at 0x9f78c0>) of role type named dpsetconstr_type
% Using role type
% Declaring dpsetconstr:(fofType->(fofType->((fofType->(fofType->Prop))->fofType)))
% FOF formula (<kernel.Constant object at 0x9f7d88>, <kernel.Sort object at 0xc80f38>) of role type named dpsetconstrI_type
% Using role type
% Declaring dpsetconstrI:Prop
% FOF formula (((eq Prop) dpsetconstrI) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((Xphi Xx) Xy)->((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu)))))))))) of role definition named dpsetconstrI
% A new definition: (((eq Prop) dpsetconstrI) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((Xphi Xx) Xy)->((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))))))))
% Defined: dpsetconstrI:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((Xphi Xx) Xy)->((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu)))))))))
% FOF formula (<kernel.Constant object at 0x9f7320>, <kernel.Sort object at 0xc80f38>) of role type named dpsetconstrSub_type
% Using role type
% Declaring dpsetconstrSub:Prop
% FOF formula (((eq Prop) dpsetconstrSub) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))), ((subset (((dpsetconstr A) B) (fun (Xx:fofType) (Xy:fofType)=> ((Xphi Xx) Xy)))) ((cartprod A) B)))) of role definition named dpsetconstrSub
% A new definition: (((eq Prop) dpsetconstrSub) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))), ((subset (((dpsetconstr A) B) (fun (Xx:fofType) (Xy:fofType)=> ((Xphi Xx) Xy)))) ((cartprod A) B))))
% Defined: dpsetconstrSub:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))), ((subset (((dpsetconstr A) B) (fun (Xx:fofType) (Xy:fofType)=> ((Xphi Xx) Xy)))) ((cartprod A) B)))
% FOF formula (<kernel.Constant object at 0x9f7b90>, <kernel.Sort object at 0xc80f38>) of role type named setOfPairsIsBReln_type
% Using role type
% Declaring setOfPairsIsBReln:Prop
% FOF formula (((eq Prop) setOfPairsIsBReln) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))), (((breln A) B) (((dpsetconstr A) B) (fun (Xx:fofType) (Xy:fofType)=> ((Xphi Xx) Xy)))))) of role definition named setOfPairsIsBReln
% A new definition: (((eq Prop) setOfPairsIsBReln) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))), (((breln A) B) (((dpsetconstr A) B) (fun (Xx:fofType) (Xy:fofType)=> ((Xphi Xx) Xy))))))
% Defined: setOfPairsIsBReln:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))), (((breln A) B) (((dpsetconstr A) B) (fun (Xx:fofType) (Xy:fofType)=> ((Xphi Xx) Xy)))))
% FOF formula (<kernel.Constant object at 0x9f75a8>, <kernel.Sort object at 0xc80f38>) of role type named dpsetconstrERa_type
% Using role type
% Declaring dpsetconstrERa:Prop
% FOF formula (((eq Prop) dpsetconstrERa) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((Xphi Xx) Xy))))))) of role definition named dpsetconstrERa
% A new definition: (((eq Prop) dpsetconstrERa) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((Xphi Xx) Xy)))))))
% Defined: dpsetconstrERa:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((Xphi Xx) Xy))))))
% FOF formula (<kernel.Constant object at 0x9f77a0>, <kernel.Sort object at 0xc80f38>) of role type named dpsetconstrEL1_type
% Using role type
% Declaring dpsetconstrEL1:Prop
% FOF formula (((eq Prop) dpsetconstrEL1) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((in Xx) A)))) of role definition named dpsetconstrEL1
% A new definition: (((eq Prop) dpsetconstrEL1) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((in Xx) A))))
% Defined: dpsetconstrEL1:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((in Xx) A)))
% FOF formula (<kernel.Constant object at 0x9f77a0>, <kernel.Sort object at 0xc80f38>) of role type named dpsetconstrEL2_type
% Using role type
% Declaring dpsetconstrEL2:Prop
% FOF formula (((eq Prop) dpsetconstrEL2) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((in Xy) B)))) of role definition named dpsetconstrEL2
% A new definition: (((eq Prop) dpsetconstrEL2) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((in Xy) B))))
% Defined: dpsetconstrEL2:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((in Xy) B)))
% FOF formula (<kernel.Constant object at 0x9f77a0>, <kernel.Sort object at 0xc80f38>) of role type named dpsetconstrER_type
% Using role type
% Declaring dpsetconstrER:Prop
% FOF formula (((eq Prop) dpsetconstrER) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((Xphi Xx) Xy)))) of role definition named dpsetconstrER
% A new definition: (((eq Prop) dpsetconstrER) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((Xphi Xx) Xy))))
% Defined: dpsetconstrER:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((Xphi Xx) Xy)))
% FOF formula (<kernel.Constant object at 0x9fb560>, <kernel.DependentProduct object at 0x9fb830>) of role type named func_type
% Using role type
% Declaring func:(fofType->(fofType->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0x9fb9e0>, <kernel.DependentProduct object at 0x9fb908>) of role type named funcSet_type
% Using role type
% Declaring funcSet:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0x9fb7a0>, <kernel.Sort object at 0xc80f38>) of role type named funcImageSingleton_type
% Using role type
% Declaring funcImageSingleton:Prop
% FOF formula (((eq Prop) funcImageSingleton) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->(singleton ((dsetconstr B) (fun (Xy:fofType)=> ((in ((kpair Xx) Xy)) Xf))))))))) of role definition named funcImageSingleton
% A new definition: (((eq Prop) funcImageSingleton) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->(singleton ((dsetconstr B) (fun (Xy:fofType)=> ((in ((kpair Xx) Xy)) Xf)))))))))
% Defined: funcImageSingleton:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->(singleton ((dsetconstr B) (fun (Xy:fofType)=> ((in ((kpair Xx) Xy)) Xf))))))))
% FOF formula (<kernel.Constant object at 0x9fb098>, <kernel.Sort object at 0xc80f38>) of role type named apProp_type
% Using role type
% Declaring apProp:Prop
% FOF formula (((eq Prop) apProp) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->((in (setunion ((dsetconstr B) (fun (Xy:fofType)=> ((in ((kpair Xx) Xy)) Xf))))) B)))))) of role definition named apProp
% A new definition: (((eq Prop) apProp) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->((in (setunion ((dsetconstr B) (fun (Xy:fofType)=> ((in ((kpair Xx) Xy)) Xf))))) B))))))
% Defined: apProp:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->((in (setunion ((dsetconstr B) (fun (Xy:fofType)=> ((in ((kpair Xx) Xy)) Xf))))) B)))))
% FOF formula (<kernel.Constant object at 0x9fb4d0>, <kernel.DependentProduct object at 0x9fb950>) of role type named ap_type
% Using role type
% Declaring ap:(fofType->(fofType->(fofType->(fofType->fofType))))
% FOF formula (<kernel.Constant object at 0x9fb8c0>, <kernel.Sort object at 0xc80f38>) of role type named app_type
% Using role type
% Declaring app:Prop
% FOF formula (((eq Prop) app) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->((in ((((ap A) B) Xf) Xx)) B)))))) of role definition named app
% A new definition: (((eq Prop) app) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->((in ((((ap A) B) Xf) Xx)) B))))))
% Defined: app:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->((in ((((ap A) B) Xf) Xx)) B)))))
% FOF formula (<kernel.Constant object at 0x9fb248>, <kernel.Sort object at 0xc80f38>) of role type named infuncsetfunc_type
% Using role type
% Declaring infuncsetfunc:Prop
% FOF formula (((eq Prop) infuncsetfunc) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(((func A) B) Xf)))) of role definition named infuncsetfunc
% A new definition: (((eq Prop) infuncsetfunc) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(((func A) B) Xf))))
% Defined: infuncsetfunc:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(((func A) B) Xf)))
% FOF formula (<kernel.Constant object at 0x9fb830>, <kernel.Sort object at 0xc80f38>) of role type named ap2p_type
% Using role type
% Declaring ap2p:Prop
% FOF formula (((eq Prop) ap2p) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->((in ((((ap A) B) Xf) Xx)) B)))))) of role definition named ap2p
% A new definition: (((eq Prop) ap2p) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->((in ((((ap A) B) Xf) Xx)) B))))))
% Defined: ap2p:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->((in ((((ap A) B) Xf) Xx)) B)))))
% FOF formula (<kernel.Constant object at 0x9fb7e8>, <kernel.Sort object at 0xc80f38>) of role type named funcinfuncset_type
% Using role type
% Declaring funcinfuncset:Prop
% FOF formula (((eq Prop) funcinfuncset) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->((in Xf) ((funcSet A) B))))) of role definition named funcinfuncset
% A new definition: (((eq Prop) funcinfuncset) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->((in Xf) ((funcSet A) B)))))
% Defined: funcinfuncset:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->((in Xf) ((funcSet A) B))))
% FOF formula (<kernel.Constant object at 0x9fb290>, <kernel.Sort object at 0xc80f38>) of role type named lamProp_type
% Using role type
% Declaring lamProp:Prop
% FOF formula (((eq Prop) lamProp) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(((func A) B) (((dpsetconstr A) B) (fun (Xx:fofType) (Xy:fofType)=> (((eq fofType) (Xf Xx)) Xy))))))) of role definition named lamProp
% A new definition: (((eq Prop) lamProp) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(((func A) B) (((dpsetconstr A) B) (fun (Xx:fofType) (Xy:fofType)=> (((eq fofType) (Xf Xx)) Xy)))))))
% Defined: lamProp:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(((func A) B) (((dpsetconstr A) B) (fun (Xx:fofType) (Xy:fofType)=> (((eq fofType) (Xf Xx)) Xy))))))
% FOF formula (<kernel.Constant object at 0x9fbb48>, <kernel.DependentProduct object at 0x9fb710>) of role type named lam_type
% Using role type
% Declaring lam:(fofType->(fofType->((fofType->fofType)->fofType)))
% FOF formula (<kernel.Constant object at 0x9fbcf8>, <kernel.Sort object at 0xc80f38>) of role type named lamp_type
% Using role type
% Declaring lamp:Prop
% FOF formula (((eq Prop) lamp) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(((func A) B) (((lam A) B) (fun (Xx:fofType)=> (Xf Xx))))))) of role definition named lamp
% A new definition: (((eq Prop) lamp) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(((func A) B) (((lam A) B) (fun (Xx:fofType)=> (Xf Xx)))))))
% Defined: lamp:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(((func A) B) (((lam A) B) (fun (Xx:fofType)=> (Xf Xx))))))
% FOF formula (<kernel.Constant object at 0x9fb170>, <kernel.Sort object at 0xc80f38>) of role type named lam2p_type
% Using role type
% Declaring lam2p:Prop
% FOF formula (((eq Prop) lam2p) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->((in (((lam A) B) (fun (Xx:fofType)=> (Xf Xx)))) ((funcSet A) B))))) of role definition named lam2p
% A new definition: (((eq Prop) lam2p) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->((in (((lam A) B) (fun (Xx:fofType)=> (Xf Xx)))) ((funcSet A) B)))))
% Defined: lam2p:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->((in (((lam A) B) (fun (Xx:fofType)=> (Xf Xx)))) ((funcSet A) B))))
% FOF formula (<kernel.Constant object at 0x9fba28>, <kernel.Sort object at 0xc80f38>) of role type named brelnall1_type
% Using role type
% Declaring brelnall1:Prop
% FOF formula (((eq Prop) brelnall1) (forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx)))))))) of role definition named brelnall1
% A new definition: (((eq Prop) brelnall1) (forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx))))))))
% Defined: brelnall1:=(forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx)))))))
% FOF formula (<kernel.Constant object at 0x9fb9e0>, <kernel.Sort object at 0xc80f38>) of role type named brelnall2_type
% Using role type
% Declaring brelnall2:Prop
% FOF formula (((eq Prop) brelnall2) (forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx)))))))) of role definition named brelnall2
% A new definition: (((eq Prop) brelnall2) (forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx))))))))
% Defined: brelnall2:=(forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx)))))))
% FOF formula (<kernel.Constant object at 0x9fbc68>, <kernel.Sort object at 0xc80f38>) of role type named ex1E2_type
% Using role type
% Declaring ex1E2:Prop
% FOF formula (((eq Prop) ex1E2) (forall (A:fofType) (Xphi:(fofType->Prop)), (((ex1 A) (fun (Xx:fofType)=> (Xphi Xx)))->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))))))) of role definition named ex1E2
% A new definition: (((eq Prop) ex1E2) (forall (A:fofType) (Xphi:(fofType->Prop)), (((ex1 A) (fun (Xx:fofType)=> (Xphi Xx)))->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy))))))))))
% Defined: ex1E2:=(forall (A:fofType) (Xphi:(fofType->Prop)), (((ex1 A) (fun (Xx:fofType)=> (Xphi Xx)))->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))))))
% FOF formula (<kernel.Constant object at 0x9fb290>, <kernel.Sort object at 0xc80f38>) of role type named funcGraphProp1_type
% Using role type
% Declaring funcGraphProp1:Prop
% FOF formula (((eq Prop) funcGraphProp1) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->((in ((kpair Xx) ((((ap A) B) Xf) Xx))) Xf)))))) of role definition named funcGraphProp1
% A new definition: (((eq Prop) funcGraphProp1) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->((in ((kpair Xx) ((((ap A) B) Xf) Xx))) Xf))))))
% Defined: funcGraphProp1:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->((in ((kpair Xx) ((((ap A) B) Xf) Xx))) Xf)))))
% FOF formula (<kernel.Constant object at 0x9fbcb0>, <kernel.Sort object at 0xc80f38>) of role type named funcGraphProp3_type
% Using role type
% Declaring funcGraphProp3:Prop
% FOF formula (((eq Prop) funcGraphProp3) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->((in ((kpair Xx) ((((ap A) B) Xf) Xx))) Xf)))))) of role definition named funcGraphProp3
% A new definition: (((eq Prop) funcGraphProp3) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->((in ((kpair Xx) ((((ap A) B) Xf) Xx))) Xf))))))
% Defined: funcGraphProp3:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->((in ((kpair Xx) ((((ap A) B) Xf) Xx))) Xf)))))
% FOF formula (<kernel.Constant object at 0x9fb518>, <kernel.Sort object at 0xc80f38>) of role type named funcGraphProp2_type
% Using role type
% Declaring funcGraphProp2:Prop
% FOF formula (((eq Prop) funcGraphProp2) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) Xf)->(((eq fofType) ((((ap A) B) Xf) Xx)) Xy))))))))) of role definition named funcGraphProp2
% A new definition: (((eq Prop) funcGraphProp2) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) Xf)->(((eq fofType) ((((ap A) B) Xf) Xx)) Xy)))))))))
% Defined: funcGraphProp2:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) Xf)->(((eq fofType) ((((ap A) B) Xf) Xx)) Xy))))))))
% FOF formula (<kernel.Constant object at 0x9fb5a8>, <kernel.Sort object at 0xc80f38>) of role type named funcextLem_type
% Using role type
% Declaring funcextLem:Prop
% FOF formula (((eq Prop) funcextLem) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xg:fofType), ((((func A) B) Xg)->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xg) Xx))))->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) Xg)->((in ((kpair Xx) Xy)) Xf)))))))))))) of role definition named funcextLem
% A new definition: (((eq Prop) funcextLem) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xg:fofType), ((((func A) B) Xg)->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xg) Xx))))->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) Xg)->((in ((kpair Xx) Xy)) Xf))))))))))))
% Defined: funcextLem:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xg:fofType), ((((func A) B) Xg)->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xg) Xx))))->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) Xg)->((in ((kpair Xx) Xy)) Xf)))))))))))
% FOF formula (<kernel.Constant object at 0x9fbab8>, <kernel.Sort object at 0xc80f38>) of role type named funcGraphProp4_type
% Using role type
% Declaring funcGraphProp4:Prop
% FOF formula (((eq Prop) funcGraphProp4) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) Xf)->(((eq fofType) ((((ap A) B) Xf) Xx)) Xy))))))))) of role definition named funcGraphProp4
% A new definition: (((eq Prop) funcGraphProp4) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) Xf)->(((eq fofType) ((((ap A) B) Xf) Xx)) Xy)))))))))
% Defined: funcGraphProp4:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) Xf)->(((eq fofType) ((((ap A) B) Xf) Xx)) Xy))))))))
% FOF formula (<kernel.Constant object at 0x9fb998>, <kernel.Sort object at 0xc80f38>) of role type named subbreln_type
% Using role type
% Declaring subbreln:Prop
% FOF formula (((eq Prop) subbreln) (forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (S:fofType), ((((breln A) B) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((subset R) S))))))) of role definition named subbreln
% A new definition: (((eq Prop) subbreln) (forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (S:fofType), ((((breln A) B) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((subset R) S)))))))
% Defined: subbreln:=(forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (S:fofType), ((((breln A) B) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((subset R) S))))))
% FOF formula (<kernel.Constant object at 0x9fb4d0>, <kernel.Sort object at 0xc80f38>) of role type named eqbreln_type
% Using role type
% Declaring eqbreln:Prop
% FOF formula (((eq Prop) eqbreln) (forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (S:fofType), ((((breln A) B) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) R))))))->(((eq fofType) R) S)))))))) of role definition named eqbreln
% A new definition: (((eq Prop) eqbreln) (forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (S:fofType), ((((breln A) B) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) R))))))->(((eq fofType) R) S))))))))
% Defined: eqbreln:=(forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (S:fofType), ((((breln A) B) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) R))))))->(((eq fofType) R) S)))))))
% FOF formula (<kernel.Constant object at 0x9fb3f8>, <kernel.Sort object at 0xc80f38>) of role type named funcext_type
% Using role type
% Declaring funcext:Prop
% FOF formula (((eq Prop) funcext) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xg:fofType), ((((func A) B) Xg)->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xg) Xx))))->(((eq fofType) Xf) Xg))))))) of role definition named funcext
% A new definition: (((eq Prop) funcext) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xg:fofType), ((((func A) B) Xg)->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xg) Xx))))->(((eq fofType) Xf) Xg)))))))
% Defined: funcext:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xg:fofType), ((((func A) B) Xg)->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xg) Xx))))->(((eq fofType) Xf) Xg))))))
% FOF formula (<kernel.Constant object at 0x9fb368>, <kernel.Sort object at 0xc80f38>) of role type named funcext2_type
% Using role type
% Declaring funcext2:Prop
% FOF formula (((eq Prop) funcext2) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xg:fofType), (((in Xg) ((funcSet A) B))->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xg) Xx))))->(((eq fofType) Xf) Xg))))))) of role definition named funcext2
% A new definition: (((eq Prop) funcext2) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xg:fofType), (((in Xg) ((funcSet A) B))->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xg) Xx))))->(((eq fofType) Xf) Xg)))))))
% Defined: funcext2:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xg:fofType), (((in Xg) ((funcSet A) B))->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xg) Xx))))->(((eq fofType) Xf) Xg))))))
% FOF formula (<kernel.Constant object at 0x9fbfc8>, <kernel.Sort object at 0xc80f38>) of role type named ap2apEq1_type
% Using role type
% Declaring ap2apEq1:Prop
% FOF formula (((eq Prop) ap2apEq1) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xf) Xx))))))) of role definition named ap2apEq1
% A new definition: (((eq Prop) ap2apEq1) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xf) Xx)))))))
% Defined: ap2apEq1:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xf) Xx))))))
% FOF formula (<kernel.Constant object at 0x9fbdd0>, <kernel.Sort object at 0xc80f38>) of role type named ap2apEq2_type
% Using role type
% Declaring ap2apEq2:Prop
% FOF formula (((eq Prop) ap2apEq2) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xf) Xx))))))) of role definition named ap2apEq2
% A new definition: (((eq Prop) ap2apEq2) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xf) Xx)))))))
% Defined: ap2apEq2:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xf) Xx))))))
% FOF formula (<kernel.Constant object at 0x9fb5f0>, <kernel.Sort object at 0xc80f38>) of role type named beta1_type
% Using role type
% Declaring beta1:Prop
% FOF formula (((eq Prop) beta1) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) (((lam A) B) (fun (Xy:fofType)=> (Xf Xy)))) Xx)) (Xf Xx))))))) of role definition named beta1
% A new definition: (((eq Prop) beta1) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) (((lam A) B) (fun (Xy:fofType)=> (Xf Xy)))) Xx)) (Xf Xx)))))))
% Defined: beta1:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) (((lam A) B) (fun (Xy:fofType)=> (Xf Xy)))) Xx)) (Xf Xx))))))
% FOF formula (<kernel.Constant object at 0x9fb5f0>, <kernel.Sort object at 0xc80f38>) of role type named eta1_type
% Using role type
% Declaring eta1:Prop
% FOF formula (((eq Prop) eta1) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(((eq fofType) (((lam A) B) (fun (Xx:fofType)=> ((((ap A) B) Xf) Xx)))) Xf)))) of role definition named eta1
% A new definition: (((eq Prop) eta1) (forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(((eq fofType) (((lam A) B) (fun (Xx:fofType)=> ((((ap A) B) Xf) Xx)))) Xf))))
% Defined: eta1:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(((eq fofType) (((lam A) B) (fun (Xx:fofType)=> ((((ap A) B) Xf) Xx)))) Xf)))
% FOF formula (<kernel.Constant object at 0x9fb5f0>, <kernel.Sort object at 0xc80f38>) of role type named lam2lamEq_type
% Using role type
% Declaring lam2lamEq:Prop
% FOF formula (((eq Prop) lam2lamEq) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(((eq fofType) (((lam A) B) (fun (Xx:fofType)=> (Xf Xx)))) (((lam A) B) (fun (Xx:fofType)=> (Xf Xx))))))) of role definition named lam2lamEq
% A new definition: (((eq Prop) lam2lamEq) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(((eq fofType) (((lam A) B) (fun (Xx:fofType)=> (Xf Xx)))) (((lam A) B) (fun (Xx:fofType)=> (Xf Xx)))))))
% Defined: lam2lamEq:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(((eq fofType) (((lam A) B) (fun (Xx:fofType)=> (Xf Xx)))) (((lam A) B) (fun (Xx:fofType)=> (Xf Xx))))))
% FOF formula (<kernel.Constant object at 0x9fe320>, <kernel.Sort object at 0xc80f38>) of role type named beta2_type
% Using role type
% Declaring beta2:Prop
% FOF formula (((eq Prop) beta2) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) (((lam A) B) (fun (Xy:fofType)=> (Xf Xy)))) Xx)) (Xf Xx))))))) of role definition named beta2
% A new definition: (((eq Prop) beta2) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) (((lam A) B) (fun (Xy:fofType)=> (Xf Xy)))) Xx)) (Xf Xx)))))))
% Defined: beta2:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) (((lam A) B) (fun (Xy:fofType)=> (Xf Xy)))) Xx)) (Xf Xx))))))
% FOF formula (<kernel.Constant object at 0x9fe638>, <kernel.Sort object at 0xc80f38>) of role type named eta2_type
% Using role type
% Declaring eta2:Prop
% FOF formula (((eq Prop) eta2) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(((eq fofType) (((lam A) B) (fun (Xx:fofType)=> ((((ap A) B) Xf) Xx)))) Xf)))) of role definition named eta2
% A new definition: (((eq Prop) eta2) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(((eq fofType) (((lam A) B) (fun (Xx:fofType)=> ((((ap A) B) Xf) Xx)))) Xf))))
% Defined: eta2:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(((eq fofType) (((lam A) B) (fun (Xx:fofType)=> ((((ap A) B) Xf) Xx)))) Xf)))
% FOF formula (<kernel.Constant object at 0x9fe998>, <kernel.Sort object at 0xc80f38>) of role type named iffalseProp1_type
% Using role type
% Declaring iffalseProp1:Prop
% FOF formula (((eq Prop) iffalseProp1) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi->False)->((in Xy) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))))))))) of role definition named iffalseProp1
% A new definition: (((eq Prop) iffalseProp1) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi->False)->((in Xy) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy))))))))))))
% Defined: iffalseProp1:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi->False)->((in Xy) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))))))))
% FOF formula (<kernel.Constant object at 0x9fe368>, <kernel.Sort object at 0xc80f38>) of role type named iffalseProp2_type
% Using role type
% Declaring iffalseProp2:Prop
% FOF formula (((eq Prop) iffalseProp2) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi->False)->(((eq fofType) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))) ((setadjoin Xy) emptyset)))))))) of role definition named iffalseProp2
% A new definition: (((eq Prop) iffalseProp2) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi->False)->(((eq fofType) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))) ((setadjoin Xy) emptyset))))))))
% Defined: iffalseProp2:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi->False)->(((eq fofType) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))) ((setadjoin Xy) emptyset)))))))
% FOF formula (<kernel.Constant object at 0x9fe7a0>, <kernel.Sort object at 0xc80f38>) of role type named iftrueProp1_type
% Using role type
% Declaring iftrueProp1:Prop
% FOF formula (((eq Prop) iftrueProp1) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(Xphi->((in Xx) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))))))))) of role definition named iftrueProp1
% A new definition: (((eq Prop) iftrueProp1) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(Xphi->((in Xx) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy))))))))))))
% Defined: iftrueProp1:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(Xphi->((in Xx) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))))))))
% FOF formula (<kernel.Constant object at 0x9fe950>, <kernel.Sort object at 0xc80f38>) of role type named iftrueProp2_type
% Using role type
% Declaring iftrueProp2:Prop
% FOF formula (((eq Prop) iftrueProp2) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(Xphi->(((eq fofType) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))) ((setadjoin Xx) emptyset)))))))) of role definition named iftrueProp2
% A new definition: (((eq Prop) iftrueProp2) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(Xphi->(((eq fofType) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))) ((setadjoin Xx) emptyset))))))))
% Defined: iftrueProp2:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(Xphi->(((eq fofType) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))) ((setadjoin Xx) emptyset)))))))
% FOF formula (<kernel.Constant object at 0x9fedd0>, <kernel.Sort object at 0xc80f38>) of role type named ifSingleton_type
% Using role type
% Declaring ifSingleton:Prop
% FOF formula (((eq Prop) ifSingleton) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(singleton ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy))))))))))) of role definition named ifSingleton
% A new definition: (((eq Prop) ifSingleton) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(singleton ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))))))))
% Defined: ifSingleton:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(singleton ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy))))))))))
% FOF formula (<kernel.Constant object at 0x9fe5a8>, <kernel.DependentProduct object at 0x9fe6c8>) of role type named if_type
% Using role type
% Declaring if:(fofType->(Prop->(fofType->(fofType->fofType))))
% FOF formula (<kernel.Constant object at 0x9fec68>, <kernel.Sort object at 0xc80f38>) of role type named ifp_type
% Using role type
% Declaring ifp:Prop
% FOF formula (((eq Prop) ifp) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((in ((((if A) Xphi) Xx) Xy)) A)))))) of role definition named ifp
% A new definition: (((eq Prop) ifp) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((in ((((if A) Xphi) Xx) Xy)) A))))))
% Defined: ifp:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((in ((((if A) Xphi) Xx) Xy)) A)))))
% FOF formula (<kernel.Constant object at 0x9fe050>, <kernel.Sort object at 0xc80f38>) of role type named theeq_type
% Using role type
% Declaring theeq:Prop
% FOF formula (((eq Prop) theeq) (forall (X:fofType), ((singleton X)->(forall (Xx:fofType), (((in Xx) X)->(((eq fofType) (setunion X)) Xx)))))) of role definition named theeq
% A new definition: (((eq Prop) theeq) (forall (X:fofType), ((singleton X)->(forall (Xx:fofType), (((in Xx) X)->(((eq fofType) (setunion X)) Xx))))))
% Defined: theeq:=(forall (X:fofType), ((singleton X)->(forall (Xx:fofType), (((in Xx) X)->(((eq fofType) (setunion X)) Xx)))))
% FOF formula (<kernel.Constant object at 0x9fe098>, <kernel.Sort object at 0xc80f38>) of role type named iftrue_type
% Using role type
% Declaring iftrue:Prop
% FOF formula (((eq Prop) iftrue) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(Xphi->(((eq fofType) ((((if A) Xphi) Xx) Xy)) Xx))))))) of role definition named iftrue
% A new definition: (((eq Prop) iftrue) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(Xphi->(((eq fofType) ((((if A) Xphi) Xx) Xy)) Xx)))))))
% Defined: iftrue:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(Xphi->(((eq fofType) ((((if A) Xphi) Xx) Xy)) Xx))))))
% FOF formula (<kernel.Constant object at 0x9fee60>, <kernel.Sort object at 0xc80f38>) of role type named iffalse_type
% Using role type
% Declaring iffalse:Prop
% FOF formula (((eq Prop) iffalse) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi->False)->(((eq fofType) ((((if A) Xphi) Xx) Xy)) Xy))))))) of role definition named iffalse
% A new definition: (((eq Prop) iffalse) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi->False)->(((eq fofType) ((((if A) Xphi) Xx) Xy)) Xy)))))))
% Defined: iffalse:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi->False)->(((eq fofType) ((((if A) Xphi) Xx) Xy)) Xy))))))
% FOF formula (<kernel.Constant object at 0x9fea70>, <kernel.Sort object at 0xc80f38>) of role type named iftrueorfalse_type
% Using role type
% Declaring iftrueorfalse:Prop
% FOF formula (((eq Prop) iftrueorfalse) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((in ((((if A) Xphi) Xx) Xy)) ((setadjoin Xx) ((setadjoin Xy) emptyset)))))))) of role definition named iftrueorfalse
% A new definition: (((eq Prop) iftrueorfalse) (forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((in ((((if A) Xphi) Xx) Xy)) ((setadjoin Xx) ((setadjoin Xy) emptyset))))))))
% Defined: iftrueorfalse:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((in ((((if A) Xphi) Xx) Xy)) ((setadjoin Xx) ((setadjoin Xy) emptyset)))))))
% FOF formula (<kernel.Constant object at 0x9feb90>, <kernel.Sort object at 0xc80f38>) of role type named binintersectT_lem_type
% Using role type
% Declaring binintersectT_lem:Prop
% FOF formula (((eq Prop) binintersectT_lem) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((binintersect X) Y)) (powerset A))))))) of role definition named binintersectT_lem
% A new definition: (((eq Prop) binintersectT_lem) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((binintersect X) Y)) (powerset A)))))))
% Defined: binintersectT_lem:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((binintersect X) Y)) (powerset A))))))
% FOF formula (<kernel.Constant object at 0x9fed40>, <kernel.Sort object at 0xc80f38>) of role type named binunionT_lem_type
% Using role type
% Declaring binunionT_lem:Prop
% FOF formula (((eq Prop) binunionT_lem) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((binunion X) Y)) (powerset A))))))) of role definition named binunionT_lem
% A new definition: (((eq Prop) binunionT_lem) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((binunion X) Y)) (powerset A)))))))
% Defined: binunionT_lem:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((binunion X) Y)) (powerset A))))))
% FOF formula (<kernel.Constant object at 0x9fedd0>, <kernel.Sort object at 0xc80f38>) of role type named powersetT_lem_type
% Using role type
% Declaring powersetT_lem:Prop
% FOF formula (((eq Prop) powersetT_lem) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->((in (powerset X)) (powerset (powerset A)))))) of role definition named powersetT_lem
% A new definition: (((eq Prop) powersetT_lem) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->((in (powerset X)) (powerset (powerset A))))))
% Defined: powersetT_lem:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->((in (powerset X)) (powerset (powerset A)))))
% FOF formula (<kernel.Constant object at 0x9fe488>, <kernel.Sort object at 0xc80f38>) of role type named setminusT_lem_type
% Using role type
% Declaring setminusT_lem:Prop
% FOF formula (((eq Prop) setminusT_lem) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus X) Y)) (powerset A))))))) of role definition named setminusT_lem
% A new definition: (((eq Prop) setminusT_lem) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus X) Y)) (powerset A)))))))
% Defined: setminusT_lem:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus X) Y)) (powerset A))))))
% FOF formula (<kernel.Constant object at 0x9fefc8>, <kernel.Sort object at 0xc80f38>) of role type named complementT_lem_type
% Using role type
% Declaring complementT_lem:Prop
% FOF formula (((eq Prop) complementT_lem) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->((in ((setminus A) X)) (powerset A))))) of role definition named complementT_lem
% A new definition: (((eq Prop) complementT_lem) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->((in ((setminus A) X)) (powerset A)))))
% Defined: complementT_lem:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->((in ((setminus A) X)) (powerset A))))
% FOF formula (<kernel.Constant object at 0x9fee18>, <kernel.Sort object at 0xc80f38>) of role type named setextT_type
% Using role type
% Declaring setextT:Prop
% FOF formula (((eq Prop) setextT) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) Y))))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) Y)->((in Xx) X))))->(((eq fofType) X) Y)))))))) of role definition named setextT
% A new definition: (((eq Prop) setextT) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) Y))))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) Y)->((in Xx) X))))->(((eq fofType) X) Y))))))))
% Defined: setextT:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) Y))))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) Y)->((in Xx) X))))->(((eq fofType) X) Y)))))))
% FOF formula (<kernel.Constant object at 0x9fe128>, <kernel.Sort object at 0xc80f38>) of role type named subsetTI_type
% Using role type
% Declaring subsetTI:Prop
% FOF formula (((eq Prop) subsetTI) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) Y))))->((subset X) Y))))))) of role definition named subsetTI
% A new definition: (((eq Prop) subsetTI) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) Y))))->((subset X) Y)))))))
% Defined: subsetTI:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) Y))))->((subset X) Y))))))
% FOF formula (<kernel.Constant object at 0x9fe830>, <kernel.Sort object at 0xc80f38>) of role type named powersetTI1_type
% Using role type
% Declaring powersetTI1:Prop
% FOF formula (((eq Prop) powersetTI1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) Y))))->((in X) (powerset Y)))))))) of role definition named powersetTI1
% A new definition: (((eq Prop) powersetTI1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) Y))))->((in X) (powerset Y))))))))
% Defined: powersetTI1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) Y))))->((in X) (powerset Y)))))))
% FOF formula (<kernel.Constant object at 0x9fe2d8>, <kernel.Sort object at 0xc80f38>) of role type named powersetTE1_type
% Using role type
% Declaring powersetTE1:Prop
% FOF formula (((eq Prop) powersetTE1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in X) (powerset Y))->(((in Xx) X)->((in Xx) Y)))))))))) of role definition named powersetTE1
% A new definition: (((eq Prop) powersetTE1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in X) (powerset Y))->(((in Xx) X)->((in Xx) Y))))))))))
% Defined: powersetTE1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in X) (powerset Y))->(((in Xx) X)->((in Xx) Y)))))))))
% FOF formula (<kernel.Constant object at 0x9fea28>, <kernel.Sort object at 0xc80f38>) of role type named complementTI1_type
% Using role type
% Declaring complementTI1:Prop
% FOF formula (((eq Prop) complementTI1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->(((in Xx) ((setminus A) X))->False))))))) of role definition named complementTI1
% A new definition: (((eq Prop) complementTI1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->(((in Xx) ((setminus A) X))->False)))))))
% Defined: complementTI1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->(((in Xx) ((setminus A) X))->False))))))
% FOF formula (<kernel.Constant object at 0x9fecb0>, <kernel.Sort object at 0xc80f38>) of role type named complementTE1_type
% Using role type
% Declaring complementTE1:Prop
% FOF formula (((eq Prop) complementTE1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) ((setminus A) X))->False)->((in Xx) X))))))) of role definition named complementTE1
% A new definition: (((eq Prop) complementTE1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) ((setminus A) X))->False)->((in Xx) X)))))))
% Defined: complementTE1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) ((setminus A) X))->False)->((in Xx) X))))))
% FOF formula (<kernel.Constant object at 0x9fe5f0>, <kernel.Sort object at 0xc80f38>) of role type named binintersectTELcontra_type
% Using role type
% Declaring binintersectTELcontra:Prop
% FOF formula (((eq Prop) binintersectTELcontra) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) X)->False)->(((in Xx) ((binintersect X) Y))->False))))))))) of role definition named binintersectTELcontra
% A new definition: (((eq Prop) binintersectTELcontra) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) X)->False)->(((in Xx) ((binintersect X) Y))->False)))))))))
% Defined: binintersectTELcontra:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) X)->False)->(((in Xx) ((binintersect X) Y))->False))))))))
% FOF formula (<kernel.Constant object at 0x9fef38>, <kernel.Sort object at 0xc80f38>) of role type named binintersectTERcontra_type
% Using role type
% Declaring binintersectTERcontra:Prop
% FOF formula (((eq Prop) binintersectTERcontra) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) Y)->False)->(((in Xx) ((binintersect X) Y))->False))))))))) of role definition named binintersectTERcontra
% A new definition: (((eq Prop) binintersectTERcontra) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) Y)->False)->(((in Xx) ((binintersect X) Y))->False)))))))))
% Defined: binintersectTERcontra:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) Y)->False)->(((in Xx) ((binintersect X) Y))->False))))))))
% FOF formula (<kernel.Constant object at 0x9fe200>, <kernel.Sort object at 0xc80f38>) of role type named contrasubsetT_type
% Using role type
% Declaring contrasubsetT:Prop
% FOF formula (((eq Prop) contrasubsetT) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((subset X) ((setminus A) Y))->(((in Xx) Y)->(((in Xx) X)->False)))))))))) of role definition named contrasubsetT
% A new definition: (((eq Prop) contrasubsetT) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((subset X) ((setminus A) Y))->(((in Xx) Y)->(((in Xx) X)->False))))))))))
% Defined: contrasubsetT:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((subset X) ((setminus A) Y))->(((in Xx) Y)->(((in Xx) X)->False)))))))))
% FOF formula (<kernel.Constant object at 0x9fe200>, <kernel.Sort object at 0xc80f38>) of role type named contrasubsetT1_type
% Using role type
% Declaring contrasubsetT1:Prop
% FOF formula (((eq Prop) contrasubsetT1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((subset X) Y)->((((in Xx) Y)->False)->(((in Xx) X)->False)))))))))) of role definition named contrasubsetT1
% A new definition: (((eq Prop) contrasubsetT1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((subset X) Y)->((((in Xx) Y)->False)->(((in Xx) X)->False))))))))))
% Defined: contrasubsetT1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((subset X) Y)->((((in Xx) Y)->False)->(((in Xx) X)->False)))))))))
% FOF formula (<kernel.Constant object at 0x9fe5f0>, <kernel.Sort object at 0xc80f38>) of role type named contrasubsetT2_type
% Using role type
% Declaring contrasubsetT2:Prop
% FOF formula (((eq Prop) contrasubsetT2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) Y)->((subset ((setminus A) Y)) ((setminus A) X)))))))) of role definition named contrasubsetT2
% A new definition: (((eq Prop) contrasubsetT2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) Y)->((subset ((setminus A) Y)) ((setminus A) X))))))))
% Defined: contrasubsetT2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) Y)->((subset ((setminus A) Y)) ((setminus A) X)))))))
% FOF formula (<kernel.Constant object at 0xa01440>, <kernel.Sort object at 0xc80f38>) of role type named contrasubsetT3_type
% Using role type
% Declaring contrasubsetT3:Prop
% FOF formula (((eq Prop) contrasubsetT3) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset ((setminus A) Y)) ((setminus A) X))->((subset X) Y))))))) of role definition named contrasubsetT3
% A new definition: (((eq Prop) contrasubsetT3) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset ((setminus A) Y)) ((setminus A) X))->((subset X) Y)))))))
% Defined: contrasubsetT3:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset ((setminus A) Y)) ((setminus A) X))->((subset X) Y))))))
% FOF formula (<kernel.Constant object at 0xa015f0>, <kernel.Sort object at 0xc80f38>) of role type named doubleComplementI1_type
% Using role type
% Declaring doubleComplementI1:Prop
% FOF formula (((eq Prop) doubleComplementI1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) ((setminus A) ((setminus A) X))))))))) of role definition named doubleComplementI1
% A new definition: (((eq Prop) doubleComplementI1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) ((setminus A) ((setminus A) X)))))))))
% Defined: doubleComplementI1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) ((setminus A) ((setminus A) X))))))))
% FOF formula (<kernel.Constant object at 0xa01710>, <kernel.Sort object at 0xc80f38>) of role type named doubleComplementE1_type
% Using role type
% Declaring doubleComplementE1:Prop
% FOF formula (((eq Prop) doubleComplementE1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((setminus A) X)))->((in Xx) X))))))) of role definition named doubleComplementE1
% A new definition: (((eq Prop) doubleComplementE1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((setminus A) X)))->((in Xx) X)))))))
% Defined: doubleComplementE1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((setminus A) X)))->((in Xx) X))))))
% FOF formula (<kernel.Constant object at 0xa01758>, <kernel.Sort object at 0xc80f38>) of role type named doubleComplementSub1_type
% Using role type
% Declaring doubleComplementSub1:Prop
% FOF formula (((eq Prop) doubleComplementSub1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->((subset X) ((setminus A) ((setminus A) X)))))) of role definition named doubleComplementSub1
% A new definition: (((eq Prop) doubleComplementSub1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->((subset X) ((setminus A) ((setminus A) X))))))
% Defined: doubleComplementSub1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->((subset X) ((setminus A) ((setminus A) X)))))
% FOF formula (<kernel.Constant object at 0xa01a28>, <kernel.Sort object at 0xc80f38>) of role type named doubleComplementSub2_type
% Using role type
% Declaring doubleComplementSub2:Prop
% FOF formula (((eq Prop) doubleComplementSub2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->((subset ((setminus A) ((setminus A) X))) X)))) of role definition named doubleComplementSub2
% A new definition: (((eq Prop) doubleComplementSub2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->((subset ((setminus A) ((setminus A) X))) X))))
% Defined: doubleComplementSub2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->((subset ((setminus A) ((setminus A) X))) X)))
% FOF formula (<kernel.Constant object at 0xa01908>, <kernel.Sort object at 0xc80f38>) of role type named doubleComplementEq_type
% Using role type
% Declaring doubleComplementEq:Prop
% FOF formula (((eq Prop) doubleComplementEq) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(((eq fofType) X) ((setminus A) ((setminus A) X)))))) of role definition named doubleComplementEq
% A new definition: (((eq Prop) doubleComplementEq) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(((eq fofType) X) ((setminus A) ((setminus A) X))))))
% Defined: doubleComplementEq:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(((eq fofType) X) ((setminus A) ((setminus A) X)))))
% FOF formula (<kernel.Constant object at 0xa01248>, <kernel.Sort object at 0xc80f38>) of role type named complementTnotintersectT_type
% Using role type
% Declaring complementTnotintersectT:Prop
% FOF formula (((eq Prop) complementTnotintersectT) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) X))->(((in Xx) ((binintersect X) Y))->False))))))))) of role definition named complementTnotintersectT
% A new definition: (((eq Prop) complementTnotintersectT) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) X))->(((in Xx) ((binintersect X) Y))->False)))))))))
% Defined: complementTnotintersectT:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) X))->(((in Xx) ((binintersect X) Y))->False))))))))
% FOF formula (<kernel.Constant object at 0xa01c20>, <kernel.Sort object at 0xc80f38>) of role type named complementImpComplementIntersect_type
% Using role type
% Declaring complementImpComplementIntersect:Prop
% FOF formula (((eq Prop) complementImpComplementIntersect) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) X))->((in Xx) ((setminus A) ((binintersect X) Y))))))))))) of role definition named complementImpComplementIntersect
% A new definition: (((eq Prop) complementImpComplementIntersect) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) X))->((in Xx) ((setminus A) ((binintersect X) Y)))))))))))
% Defined: complementImpComplementIntersect:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) X))->((in Xx) ((setminus A) ((binintersect X) Y))))))))))
% FOF formula (<kernel.Constant object at 0xa018c0>, <kernel.Sort object at 0xc80f38>) of role type named complementSubsetComplementIntersect_type
% Using role type
% Declaring complementSubsetComplementIntersect:Prop
% FOF formula (((eq Prop) complementSubsetComplementIntersect) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((subset ((setminus A) X)) ((setminus A) ((binintersect X) Y)))))))) of role definition named complementSubsetComplementIntersect
% A new definition: (((eq Prop) complementSubsetComplementIntersect) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((subset ((setminus A) X)) ((setminus A) ((binintersect X) Y))))))))
% Defined: complementSubsetComplementIntersect:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((subset ((setminus A) X)) ((setminus A) ((binintersect X) Y)))))))
% FOF formula (<kernel.Constant object at 0xa016c8>, <kernel.Sort object at 0xc80f38>) of role type named complementInPowersetComplementIntersect_type
% Using role type
% Declaring complementInPowersetComplementIntersect:Prop
% FOF formula (((eq Prop) complementInPowersetComplementIntersect) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) X)) (powerset ((setminus A) ((binintersect X) Y))))))))) of role definition named complementInPowersetComplementIntersect
% A new definition: (((eq Prop) complementInPowersetComplementIntersect) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) X)) (powerset ((setminus A) ((binintersect X) Y)))))))))
% Defined: complementInPowersetComplementIntersect:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) X)) (powerset ((setminus A) ((binintersect X) Y))))))))
% FOF formula (<kernel.Constant object at 0xa018c0>, <kernel.Sort object at 0xc80f38>) of role type named contraSubsetComplement_type
% Using role type
% Declaring contraSubsetComplement:Prop
% FOF formula (((eq Prop) contraSubsetComplement) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) ((setminus A) Y))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) Y)->((in Xx) ((setminus A) X))))))))))) of role definition named contraSubsetComplement
% A new definition: (((eq Prop) contraSubsetComplement) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) ((setminus A) Y))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) Y)->((in Xx) ((setminus A) X)))))))))))
% Defined: contraSubsetComplement:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) ((setminus A) Y))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) Y)->((in Xx) ((setminus A) X))))))))))
% FOF formula (<kernel.Constant object at 0xa01b00>, <kernel.Sort object at 0xc80f38>) of role type named complementTcontraSubset_type
% Using role type
% Declaring complementTcontraSubset:Prop
% FOF formula (((eq Prop) complementTcontraSubset) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) ((setminus A) Y))->((subset Y) ((setminus A) X)))))))) of role definition named complementTcontraSubset
% A new definition: (((eq Prop) complementTcontraSubset) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) ((setminus A) Y))->((subset Y) ((setminus A) X))))))))
% Defined: complementTcontraSubset:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) ((setminus A) Y))->((subset Y) ((setminus A) X)))))))
% FOF formula (<kernel.Constant object at 0xa01c68>, <kernel.Sort object at 0xc80f38>) of role type named binunionTILcontra_type
% Using role type
% Declaring binunionTILcontra:Prop
% FOF formula (((eq Prop) binunionTILcontra) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) ((binunion X) Y))->False)->(((in Xx) X)->False))))))))) of role definition named binunionTILcontra
% A new definition: (((eq Prop) binunionTILcontra) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) ((binunion X) Y))->False)->(((in Xx) X)->False)))))))))
% Defined: binunionTILcontra:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) ((binunion X) Y))->False)->(((in Xx) X)->False))))))))
% FOF formula (<kernel.Constant object at 0xa01b00>, <kernel.Sort object at 0xc80f38>) of role type named binunionTIRcontra_type
% Using role type
% Declaring binunionTIRcontra:Prop
% FOF formula (((eq Prop) binunionTIRcontra) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) ((binunion X) Y))->False)->(((in Xx) Y)->False))))))))) of role definition named binunionTIRcontra
% A new definition: (((eq Prop) binunionTIRcontra) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) ((binunion X) Y))->False)->(((in Xx) Y)->False)))))))))
% Defined: binunionTIRcontra:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) ((binunion X) Y))->False)->(((in Xx) Y)->False))))))))
% FOF formula (<kernel.Constant object at 0xa01ef0>, <kernel.Sort object at 0xc80f38>) of role type named inIntersectImpInUnion_type
% Using role type
% Declaring inIntersectImpInUnion:Prop
% FOF formula (((eq Prop) inIntersectImpInUnion) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect X) Y))->((in Xx) ((binunion X) Z)))))))))))) of role definition named inIntersectImpInUnion
% A new definition: (((eq Prop) inIntersectImpInUnion) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect X) Y))->((in Xx) ((binunion X) Z))))))))))))
% Defined: inIntersectImpInUnion:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect X) Y))->((in Xx) ((binunion X) Z)))))))))))
% FOF formula (<kernel.Constant object at 0xa01128>, <kernel.Sort object at 0xc80f38>) of role type named inIntersectImpInUnion2_type
% Using role type
% Declaring inIntersectImpInUnion2:Prop
% FOF formula (((eq Prop) inIntersectImpInUnion2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect X) Y))->((in Xx) ((binunion Y) Z)))))))))))) of role definition named inIntersectImpInUnion2
% A new definition: (((eq Prop) inIntersectImpInUnion2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect X) Y))->((in Xx) ((binunion Y) Z))))))))))))
% Defined: inIntersectImpInUnion2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect X) Y))->((in Xx) ((binunion Y) Z)))))))))))
% FOF formula (<kernel.Constant object at 0xa01128>, <kernel.Sort object at 0xc80f38>) of role type named inIntersectImpInIntersectUnions_type
% Using role type
% Declaring inIntersectImpInIntersectUnions:Prop
% FOF formula (((eq Prop) inIntersectImpInIntersectUnions) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect X) Y))->((in Xx) ((binintersect ((binunion X) Z)) ((binunion Y) Z))))))))))))) of role definition named inIntersectImpInIntersectUnions
% A new definition: (((eq Prop) inIntersectImpInIntersectUnions) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect X) Y))->((in Xx) ((binintersect ((binunion X) Z)) ((binunion Y) Z)))))))))))))
% Defined: inIntersectImpInIntersectUnions:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect X) Y))->((in Xx) ((binintersect ((binunion X) Z)) ((binunion Y) Z))))))))))))
% FOF formula (<kernel.Constant object at 0xa014d0>, <kernel.Sort object at 0xc80f38>) of role type named intersectInPowersetIntersectUnions_type
% Using role type
% Declaring intersectInPowersetIntersectUnions:Prop
% FOF formula (((eq Prop) intersectInPowersetIntersectUnions) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->((in ((binintersect X) Y)) (powerset ((binintersect ((binunion X) Z)) ((binunion Y) Z))))))))))) of role definition named intersectInPowersetIntersectUnions
% A new definition: (((eq Prop) intersectInPowersetIntersectUnions) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->((in ((binintersect X) Y)) (powerset ((binintersect ((binunion X) Z)) ((binunion Y) Z)))))))))))
% Defined: intersectInPowersetIntersectUnions:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->((in ((binintersect X) Y)) (powerset ((binintersect ((binunion X) Z)) ((binunion Y) Z))))))))))
% FOF formula (<kernel.Constant object at 0xa01e18>, <kernel.Sort object at 0xc80f38>) of role type named inComplementUnionImpNotIn1_type
% Using role type
% Declaring inComplementUnionImpNotIn1:Prop
% FOF formula (((eq Prop) inComplementUnionImpNotIn1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->(((in Xx) X)->False))))))))) of role definition named inComplementUnionImpNotIn1
% A new definition: (((eq Prop) inComplementUnionImpNotIn1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->(((in Xx) X)->False)))))))))
% Defined: inComplementUnionImpNotIn1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->(((in Xx) X)->False))))))))
% FOF formula (<kernel.Constant object at 0xa014d0>, <kernel.Sort object at 0xc80f38>) of role type named inComplementUnionImpInComplement1_type
% Using role type
% Declaring inComplementUnionImpInComplement1:Prop
% FOF formula (((eq Prop) inComplementUnionImpInComplement1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((setminus A) X)))))))))) of role definition named inComplementUnionImpInComplement1
% A new definition: (((eq Prop) inComplementUnionImpInComplement1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((setminus A) X))))))))))
% Defined: inComplementUnionImpInComplement1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((setminus A) X)))))))))
% FOF formula (<kernel.Constant object at 0xa01bd8>, <kernel.Sort object at 0xc80f38>) of role type named binunionTE_type
% Using role type
% Declaring binunionTE:Prop
% FOF formula (((eq Prop) binunionTE) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(((in Xx) ((binunion X) Y))->((((in Xx) X)->Xphi)->((((in Xx) Y)->Xphi)->Xphi)))))))))) of role definition named binunionTE
% A new definition: (((eq Prop) binunionTE) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(((in Xx) ((binunion X) Y))->((((in Xx) X)->Xphi)->((((in Xx) Y)->Xphi)->Xphi))))))))))
% Defined: binunionTE:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(((in Xx) ((binunion X) Y))->((((in Xx) X)->Xphi)->((((in Xx) Y)->Xphi)->Xphi)))))))))
% FOF formula (<kernel.Constant object at 0xa014d0>, <kernel.Sort object at 0xc80f38>) of role type named binunionTEcontra_type
% Using role type
% Declaring binunionTEcontra:Prop
% FOF formula (((eq Prop) binunionTEcontra) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) X)->False)->((((in Xx) Y)->False)->(((in Xx) ((binunion X) Y))->False)))))))))) of role definition named binunionTEcontra
% A new definition: (((eq Prop) binunionTEcontra) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) X)->False)->((((in Xx) Y)->False)->(((in Xx) ((binunion X) Y))->False))))))))))
% Defined: binunionTEcontra:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) X)->False)->((((in Xx) Y)->False)->(((in Xx) ((binunion X) Y))->False)))))))))
% FOF formula (<kernel.Constant object at 0xa01e18>, <kernel.Sort object at 0xc80f38>) of role type named demorgan2a1_type
% Using role type
% Declaring demorgan2a1:Prop
% FOF formula (((eq Prop) demorgan2a1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((setminus A) X)))))))))) of role definition named demorgan2a1
% A new definition: (((eq Prop) demorgan2a1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((setminus A) X))))))))))
% Defined: demorgan2a1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((setminus A) X)))))))))
% FOF formula (<kernel.Constant object at 0xa014d0>, <kernel.Sort object at 0xc80f38>) of role type named complementUnionInPowersetComplement_type
% Using role type
% Declaring complementUnionInPowersetComplement:Prop
% FOF formula (((eq Prop) complementUnionInPowersetComplement) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) ((binunion X) Y))) (powerset ((setminus A) X)))))))) of role definition named complementUnionInPowersetComplement
% A new definition: (((eq Prop) complementUnionInPowersetComplement) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) ((binunion X) Y))) (powerset ((setminus A) X))))))))
% Defined: complementUnionInPowersetComplement:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) ((binunion X) Y))) (powerset ((setminus A) X)))))))
% FOF formula (<kernel.Constant object at 0xa01e18>, <kernel.Sort object at 0xc80f38>) of role type named demorgan2a2_type
% Using role type
% Declaring demorgan2a2:Prop
% FOF formula (((eq Prop) demorgan2a2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((setminus A) Y)))))))))) of role definition named demorgan2a2
% A new definition: (((eq Prop) demorgan2a2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((setminus A) Y))))))))))
% Defined: demorgan2a2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((setminus A) Y)))))))))
% FOF formula (<kernel.Constant object at 0xa041b8>, <kernel.Sort object at 0xc80f38>) of role type named demorgan1a_type
% Using role type
% Declaring demorgan1a:Prop
% FOF formula (((eq Prop) demorgan1a) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binintersect X) Y)))->((in Xx) ((binunion ((setminus A) X)) ((setminus A) Y))))))))))) of role definition named demorgan1a
% A new definition: (((eq Prop) demorgan1a) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binintersect X) Y)))->((in Xx) ((binunion ((setminus A) X)) ((setminus A) Y)))))))))))
% Defined: demorgan1a:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binintersect X) Y)))->((in Xx) ((binunion ((setminus A) X)) ((setminus A) Y))))))))))
% FOF formula (<kernel.Constant object at 0xa04998>, <kernel.Sort object at 0xc80f38>) of role type named demorgan1b_type
% Using role type
% Declaring demorgan1b:Prop
% FOF formula (((eq Prop) demorgan1b) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binunion ((setminus A) X)) ((setminus A) Y)))->((in Xx) ((setminus A) ((binintersect X) Y))))))))))) of role definition named demorgan1b
% A new definition: (((eq Prop) demorgan1b) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binunion ((setminus A) X)) ((setminus A) Y)))->((in Xx) ((setminus A) ((binintersect X) Y)))))))))))
% Defined: demorgan1b:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binunion ((setminus A) X)) ((setminus A) Y)))->((in Xx) ((setminus A) ((binintersect X) Y))))))))))
% FOF formula (<kernel.Constant object at 0xa047e8>, <kernel.Sort object at 0xc80f38>) of role type named demorgan1_type
% Using role type
% Declaring demorgan1:Prop
% FOF formula (((eq Prop) demorgan1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((eq fofType) ((setminus A) ((binintersect X) Y))) ((binunion ((setminus A) X)) ((setminus A) Y)))))))) of role definition named demorgan1
% A new definition: (((eq Prop) demorgan1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((eq fofType) ((setminus A) ((binintersect X) Y))) ((binunion ((setminus A) X)) ((setminus A) Y))))))))
% Defined: demorgan1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((eq fofType) ((setminus A) ((binintersect X) Y))) ((binunion ((setminus A) X)) ((setminus A) Y)))))))
% FOF formula (<kernel.Constant object at 0xa04680>, <kernel.Sort object at 0xc80f38>) of role type named demorgan2a_type
% Using role type
% Declaring demorgan2a:Prop
% FOF formula (((eq Prop) demorgan2a) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((binintersect ((setminus A) X)) ((setminus A) Y))))))))))) of role definition named demorgan2a
% A new definition: (((eq Prop) demorgan2a) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((binintersect ((setminus A) X)) ((setminus A) Y)))))))))))
% Defined: demorgan2a:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((binintersect ((setminus A) X)) ((setminus A) Y))))))))))
% FOF formula (<kernel.Constant object at 0xa04878>, <kernel.Sort object at 0xc80f38>) of role type named demorgan2b2_type
% Using role type
% Declaring demorgan2b2:Prop
% FOF formula (((eq Prop) demorgan2b2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) X))->(((in Xx) ((setminus A) Y))->((in Xx) ((setminus A) ((binunion X) Y)))))))))))) of role definition named demorgan2b2
% A new definition: (((eq Prop) demorgan2b2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) X))->(((in Xx) ((setminus A) Y))->((in Xx) ((setminus A) ((binunion X) Y))))))))))))
% Defined: demorgan2b2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) X))->(((in Xx) ((setminus A) Y))->((in Xx) ((setminus A) ((binunion X) Y)))))))))))
% FOF formula (<kernel.Constant object at 0xa045f0>, <kernel.Sort object at 0xc80f38>) of role type named demorgan2b_type
% Using role type
% Declaring demorgan2b:Prop
% FOF formula (((eq Prop) demorgan2b) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect ((setminus A) X)) ((setminus A) Y)))->((in Xx) ((setminus A) ((binunion X) Y))))))))))) of role definition named demorgan2b
% A new definition: (((eq Prop) demorgan2b) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect ((setminus A) X)) ((setminus A) Y)))->((in Xx) ((setminus A) ((binunion X) Y)))))))))))
% Defined: demorgan2b:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect ((setminus A) X)) ((setminus A) Y)))->((in Xx) ((setminus A) ((binunion X) Y))))))))))
% FOF formula (<kernel.Constant object at 0xa04c68>, <kernel.Sort object at 0xc80f38>) of role type named demorgan2_type
% Using role type
% Declaring demorgan2:Prop
% FOF formula (((eq Prop) demorgan2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((eq fofType) ((setminus A) ((binunion X) Y))) ((binintersect ((setminus A) X)) ((setminus A) Y)))))))) of role definition named demorgan2
% A new definition: (((eq Prop) demorgan2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((eq fofType) ((setminus A) ((binunion X) Y))) ((binintersect ((setminus A) X)) ((setminus A) Y))))))))
% Defined: demorgan2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((eq fofType) ((setminus A) ((binunion X) Y))) ((binintersect ((setminus A) X)) ((setminus A) Y)))))))
% FOF formula (<kernel.Constant object at 0xa04290>, <kernel.Sort object at 0xc80f38>) of role type named woz13rule0_type
% Using role type
% Declaring woz13rule0:Prop
% FOF formula (((eq Prop) woz13rule0) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) ((binintersect X) Y))->((in Xx) A)))))))) of role definition named woz13rule0
% A new definition: (((eq Prop) woz13rule0) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) ((binintersect X) Y))->((in Xx) A))))))))
% Defined: woz13rule0:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) ((binintersect X) Y))->((in Xx) A)))))))
% FOF formula (<kernel.Constant object at 0xa04d88>, <kernel.Sort object at 0xc80f38>) of role type named woz13rule1_type
% Using role type
% Declaring woz13rule1:Prop
% FOF formula (((eq Prop) woz13rule1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(((subset X) Z)->((subset ((binintersect X) Y)) Z))))))))) of role definition named woz13rule1
% A new definition: (((eq Prop) woz13rule1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(((subset X) Z)->((subset ((binintersect X) Y)) Z)))))))))
% Defined: woz13rule1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(((subset X) Z)->((subset ((binintersect X) Y)) Z))))))))
% FOF formula (<kernel.Constant object at 0xa04758>, <kernel.Sort object at 0xc80f38>) of role type named woz13rule2_type
% Using role type
% Declaring woz13rule2:Prop
% FOF formula (((eq Prop) woz13rule2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(((subset Y) Z)->((subset ((binintersect X) Y)) Z))))))))) of role definition named woz13rule2
% A new definition: (((eq Prop) woz13rule2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(((subset Y) Z)->((subset ((binintersect X) Y)) Z)))))))))
% Defined: woz13rule2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(((subset Y) Z)->((subset ((binintersect X) Y)) Z))))))))
% FOF formula (<kernel.Constant object at 0xa04dd0>, <kernel.Sort object at 0xc80f38>) of role type named woz13rule3_type
% Using role type
% Declaring woz13rule3:Prop
% FOF formula (((eq Prop) woz13rule3) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(((subset X) Y)->(((subset X) Z)->((subset X) ((binintersect Y) Z))))))))))) of role definition named woz13rule3
% A new definition: (((eq Prop) woz13rule3) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(((subset X) Y)->(((subset X) Z)->((subset X) ((binintersect Y) Z)))))))))))
% Defined: woz13rule3:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(((subset X) Y)->(((subset X) Z)->((subset X) ((binintersect Y) Z))))))))))
% FOF formula (<kernel.Constant object at 0xa04368>, <kernel.Sort object at 0xc80f38>) of role type named woz13rule4_type
% Using role type
% Declaring woz13rule4:Prop
% FOF formula (((eq Prop) woz13rule4) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (W:fofType), (((in W) (powerset A))->(((subset X) Z)->(((subset Y) W)->((subset ((binintersect X) Y)) ((binintersect Z) W))))))))))))) of role definition named woz13rule4
% A new definition: (((eq Prop) woz13rule4) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (W:fofType), (((in W) (powerset A))->(((subset X) Z)->(((subset Y) W)->((subset ((binintersect X) Y)) ((binintersect Z) W)))))))))))))
% Defined: woz13rule4:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (W:fofType), (((in W) (powerset A))->(((subset X) Z)->(((subset Y) W)->((subset ((binintersect X) Y)) ((binintersect Z) W))))))))))))
% FOF formula (<kernel.Constant object at 0xa04cf8>, <kernel.Sort object at 0xc80f38>) of role type named woz1_1_type
% Using role type
% Declaring woz1_1:Prop
% FOF formula (((eq Prop) woz1_1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) X)) (powerset ((setminus A) ((binintersect X) Y))))))))) of role definition named woz1_1
% A new definition: (((eq Prop) woz1_1) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) X)) (powerset ((setminus A) ((binintersect X) Y)))))))))
% Defined: woz1_1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) X)) (powerset ((setminus A) ((binintersect X) Y))))))))
% FOF formula (<kernel.Constant object at 0xa04e18>, <kernel.Sort object at 0xc80f38>) of role type named woz1_2_type
% Using role type
% Declaring woz1_2:Prop
% FOF formula (((eq Prop) woz1_2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (W:fofType), (((in W) (powerset A))->(((eq fofType) ((setminus A) ((binintersect ((binunion X) Y)) ((binunion Z) W)))) ((binunion ((binintersect ((setminus A) X)) ((setminus A) Y))) ((binintersect ((setminus A) Z)) ((setminus A) W))))))))))))) of role definition named woz1_2
% A new definition: (((eq Prop) woz1_2) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (W:fofType), (((in W) (powerset A))->(((eq fofType) ((setminus A) ((binintersect ((binunion X) Y)) ((binunion Z) W)))) ((binunion ((binintersect ((setminus A) X)) ((setminus A) Y))) ((binintersect ((setminus A) Z)) ((setminus A) W)))))))))))))
% Defined: woz1_2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (W:fofType), (((in W) (powerset A))->(((eq fofType) ((setminus A) ((binintersect ((binunion X) Y)) ((binunion Z) W)))) ((binunion ((binintersect ((setminus A) X)) ((setminus A) Y))) ((binintersect ((setminus A) Z)) ((setminus A) W))))))))))))
% FOF formula (<kernel.Constant object at 0xa04638>, <kernel.Sort object at 0xc80f38>) of role type named woz1_3_type
% Using role type
% Declaring woz1_3:Prop
% FOF formula (((eq Prop) woz1_3) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->((in ((binintersect X) Y)) (powerset ((binintersect ((binunion X) Z)) ((binunion Y) Z))))))))))) of role definition named woz1_3
% A new definition: (((eq Prop) woz1_3) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->((in ((binintersect X) Y)) (powerset ((binintersect ((binunion X) Z)) ((binunion Y) Z)))))))))))
% Defined: woz1_3:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->((in ((binintersect X) Y)) (powerset ((binintersect ((binunion X) Z)) ((binunion Y) Z))))))))))
% FOF formula (<kernel.Constant object at 0xa043b0>, <kernel.Sort object at 0xc80f38>) of role type named woz1_4_type
% Using role type
% Declaring woz1_4:Prop
% FOF formula (((eq Prop) woz1_4) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) ((setminus A) Y))->((subset Y) ((setminus A) X)))))))) of role definition named woz1_4
% A new definition: (((eq Prop) woz1_4) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) ((setminus A) Y))->((subset Y) ((setminus A) X))))))))
% Defined: woz1_4:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) ((setminus A) Y))->((subset Y) ((setminus A) X)))))))
% FOF formula (<kernel.Constant object at 0xa048c0>, <kernel.Sort object at 0xc80f38>) of role type named woz1_5_type
% Using role type
% Declaring woz1_5:Prop
% FOF formula (((eq Prop) woz1_5) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) ((binunion X) Y))) (powerset ((setminus A) X)))))))) of role definition named woz1_5
% A new definition: (((eq Prop) woz1_5) (forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) ((binunion X) Y))) (powerset ((setminus A) X))))))))
% Defined: woz1_5:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) ((binunion X) Y))) (powerset ((setminus A) X)))))))
% FOF formula (<kernel.Constant object at 0xa04b48>, <kernel.DependentProduct object at 0xa04a28>) of role type named breln1_type
% Using role type
% Declaring breln1:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xa04710>, <kernel.Sort object at 0xc80f38>) of role type named breln1all2_type
% Using role type
% Declaring breln1all2:Prop
% FOF formula (((eq Prop) breln1all2) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx)))))))) of role definition named breln1all2
% A new definition: (((eq Prop) breln1all2) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx))))))))
% Defined: breln1all2:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx)))))))
% FOF formula (<kernel.Constant object at 0xa048c0>, <kernel.DependentProduct object at 0xa046c8>) of role type named breln1Set_type
% Using role type
% Declaring breln1Set:(fofType->fofType)
% FOF formula (<kernel.Constant object at 0xa04b48>, <kernel.Sort object at 0xc80f38>) of role type named breln1SetBreln1_type
% Using role type
% Declaring breln1SetBreln1:Prop
% FOF formula (((eq Prop) breln1SetBreln1) (forall (A:fofType) (R:fofType), (((in R) (breln1Set A))->((breln1 A) R)))) of role definition named breln1SetBreln1
% A new definition: (((eq Prop) breln1SetBreln1) (forall (A:fofType) (R:fofType), (((in R) (breln1Set A))->((breln1 A) R))))
% Defined: breln1SetBreln1:=(forall (A:fofType) (R:fofType), (((in R) (breln1Set A))->((breln1 A) R)))
% FOF formula (<kernel.Constant object at 0xa04710>, <kernel.DependentProduct object at 0xa04ea8>) of role type named transitive_type
% Using role type
% Declaring transitive:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xa04cb0>, <kernel.DependentProduct object at 0xa04320>) of role type named antisymmetric_type
% Using role type
% Declaring antisymmetric:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xa04b48>, <kernel.DependentProduct object at 0xa04710>) of role type named reflexive_type
% Using role type
% Declaring reflexive:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xa048c0>, <kernel.DependentProduct object at 0xa04cb0>) of role type named refltransitive_type
% Using role type
% Declaring refltransitive:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xa04fc8>, <kernel.DependentProduct object at 0xa04b48>) of role type named refllinearorder_type
% Using role type
% Declaring refllinearorder:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xa04ea8>, <kernel.DependentProduct object at 0xa048c0>) of role type named reflwellordering_type
% Using role type
% Declaring reflwellordering:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xa04320>, <kernel.Sort object at 0xc80f38>) of role type named choice2fnsingleton_type
% Using role type
% Declaring choice2fnsingleton:Prop
% FOF formula (((eq Prop) choice2fnsingleton) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))), ((forall (Xx:fofType), (((in Xx) A)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) B)) ((Xphi Xx) Xy))))))->(forall (R:fofType), (((in R) (breln1Set B))->(((reflwellordering B) R)->(forall (Xx:fofType), (((in Xx) A)->(singleton ((dsetconstr B) (fun (Xy:fofType)=> ((and ((Xphi Xx) Xy)) (forall (Xz:fofType), (((in Xz) B)->(((Xphi Xx) Xz)->((in ((kpair Xy) Xz)) R)))))))))))))))) of role definition named choice2fnsingleton
% A new definition: (((eq Prop) choice2fnsingleton) (forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))), ((forall (Xx:fofType), (((in Xx) A)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) B)) ((Xphi Xx) Xy))))))->(forall (R:fofType), (((in R) (breln1Set B))->(((reflwellordering B) R)->(forall (Xx:fofType), (((in Xx) A)->(singleton ((dsetconstr B) (fun (Xy:fofType)=> ((and ((Xphi Xx) Xy)) (forall (Xz:fofType), (((in Xz) B)->(((Xphi Xx) Xz)->((in ((kpair Xy) Xz)) R))))))))))))))))
% Defined: choice2fnsingleton:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))), ((forall (Xx:fofType), (((in Xx) A)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) B)) ((Xphi Xx) Xy))))))->(forall (R:fofType), (((in R) (breln1Set B))->(((reflwellordering B) R)->(forall (Xx:fofType), (((in Xx) A)->(singleton ((dsetconstr B) (fun (Xy:fofType)=> ((and ((Xphi Xx) Xy)) (forall (Xz:fofType), (((in Xz) B)->(((Xphi Xx) Xz)->((in ((kpair Xy) Xz)) R)))))))))))))))
% FOF formula (<kernel.Constant object at 0xa04710>, <kernel.Sort object at 0xc80f38>) of role type named setOfPairsIsBReln1_type
% Using role type
% Declaring setOfPairsIsBReln1:Prop
% FOF formula (((eq Prop) setOfPairsIsBReln1) (forall (A:fofType) (Xphi:(fofType->(fofType->Prop))), ((breln1 A) (((dpsetconstr A) A) (fun (Xx:fofType) (Xy:fofType)=> ((Xphi Xx) Xy)))))) of role definition named setOfPairsIsBReln1
% A new definition: (((eq Prop) setOfPairsIsBReln1) (forall (A:fofType) (Xphi:(fofType->(fofType->Prop))), ((breln1 A) (((dpsetconstr A) A) (fun (Xx:fofType) (Xy:fofType)=> ((Xphi Xx) Xy))))))
% Defined: setOfPairsIsBReln1:=(forall (A:fofType) (Xphi:(fofType->(fofType->Prop))), ((breln1 A) (((dpsetconstr A) A) (fun (Xx:fofType) (Xy:fofType)=> ((Xphi Xx) Xy)))))
% FOF formula (<kernel.Constant object at 0xa04fc8>, <kernel.Sort object at 0xc80f38>) of role type named breln1all1_type
% Using role type
% Declaring breln1all1:Prop
% FOF formula (((eq Prop) breln1all1) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx)))))))) of role definition named breln1all1
% A new definition: (((eq Prop) breln1all1) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx))))))))
% Defined: breln1all1:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx)))))))
% FOF formula (<kernel.Constant object at 0xa04fc8>, <kernel.Sort object at 0xc80f38>) of role type named subbreln1_type
% Using role type
% Declaring subbreln1:Prop
% FOF formula (((eq Prop) subbreln1) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((subset R) S))))))) of role definition named subbreln1
% A new definition: (((eq Prop) subbreln1) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((subset R) S)))))))
% Defined: subbreln1:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((subset R) S))))))
% FOF formula (<kernel.Constant object at 0xa04320>, <kernel.Sort object at 0xc80f38>) of role type named eqbreln1_type
% Using role type
% Declaring eqbreln1:Prop
% FOF formula (((eq Prop) eqbreln1) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) R))))))->(((eq fofType) R) S)))))))) of role definition named eqbreln1
% A new definition: (((eq Prop) eqbreln1) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) R))))))->(((eq fofType) R) S))))))))
% Defined: eqbreln1:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) R))))))->(((eq fofType) R) S)))))))
% FOF formula (<kernel.Constant object at 0xa067a0>, <kernel.DependentProduct object at 0xa06170>) of role type named breln1invset_type
% Using role type
% Declaring breln1invset:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0xa06320>, <kernel.Sort object at 0xc80f38>) of role type named breln1invprop_type
% Using role type
% Declaring breln1invprop:Prop
% FOF formula (((eq Prop) breln1invprop) (forall (A:fofType) (R:fofType), (((breln1 A) R)->((breln1 A) ((breln1invset A) R))))) of role definition named breln1invprop
% A new definition: (((eq Prop) breln1invprop) (forall (A:fofType) (R:fofType), (((breln1 A) R)->((breln1 A) ((breln1invset A) R)))))
% Defined: breln1invprop:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->((breln1 A) ((breln1invset A) R))))
% FOF formula (<kernel.Constant object at 0xa06200>, <kernel.Sort object at 0xc80f38>) of role type named breln1invI_type
% Using role type
% Declaring breln1invI:Prop
% FOF formula (((eq Prop) breln1invI) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xy) Xx)) ((breln1invset A) R)))))))))) of role definition named breln1invI
% A new definition: (((eq Prop) breln1invI) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xy) Xx)) ((breln1invset A) R))))))))))
% Defined: breln1invI:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xy) Xx)) ((breln1invset A) R)))))))))
% FOF formula (<kernel.Constant object at 0xa067a0>, <kernel.Sort object at 0xc80f38>) of role type named breln1invE_type
% Using role type
% Declaring breln1invE:Prop
% FOF formula (((eq Prop) breln1invE) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xy) Xx)) ((breln1invset A) R))->((in ((kpair Xx) Xy)) R))))))))) of role definition named breln1invE
% A new definition: (((eq Prop) breln1invE) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xy) Xx)) ((breln1invset A) R))->((in ((kpair Xx) Xy)) R)))))))))
% Defined: breln1invE:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xy) Xx)) ((breln1invset A) R))->((in ((kpair Xx) Xy)) R))))))))
% FOF formula (<kernel.Constant object at 0xa066c8>, <kernel.DependentProduct object at 0xa06050>) of role type named breln1compset_type
% Using role type
% Declaring breln1compset:(fofType->(fofType->(fofType->fofType)))
% FOF formula (<kernel.Constant object at 0xa069e0>, <kernel.Sort object at 0xc80f38>) of role type named breln1compprop_type
% Using role type
% Declaring breln1compprop:Prop
% FOF formula (((eq Prop) breln1compprop) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((breln1 A) (((breln1compset A) R) S))))))) of role definition named breln1compprop
% A new definition: (((eq Prop) breln1compprop) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((breln1 A) (((breln1compset A) R) S)))))))
% Defined: breln1compprop:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((breln1 A) (((breln1compset A) R) S))))))
% FOF formula (<kernel.Constant object at 0xa06878>, <kernel.Sort object at 0xc80f38>) of role type named breln1compI_type
% Using role type
% Declaring breln1compI:Prop
% FOF formula (((eq Prop) breln1compI) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(forall (Xz:fofType), (((in Xz) A)->(((in ((kpair Xx) Xz)) R)->(((in ((kpair Xz) Xy)) S)->((in ((kpair Xx) Xy)) (((breln1compset A) R) S))))))))))))))) of role definition named breln1compI
% A new definition: (((eq Prop) breln1compI) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(forall (Xz:fofType), (((in Xz) A)->(((in ((kpair Xx) Xz)) R)->(((in ((kpair Xz) Xy)) S)->((in ((kpair Xx) Xy)) (((breln1compset A) R) S)))))))))))))))
% Defined: breln1compI:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(forall (Xz:fofType), (((in Xz) A)->(((in ((kpair Xx) Xz)) R)->(((in ((kpair Xz) Xy)) S)->((in ((kpair Xx) Xy)) (((breln1compset A) R) S))))))))))))))
% FOF formula (<kernel.Constant object at 0xa06758>, <kernel.Sort object at 0xc80f38>) of role type named breln1compE_type
% Using role type
% Declaring breln1compE:Prop
% FOF formula (((eq Prop) breln1compE) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) (((breln1compset A) R) S))->((ex fofType) (fun (Xz:fofType)=> ((and ((and ((in Xz) A)) ((in ((kpair Xx) Xz)) R))) ((in ((kpair Xz) Xy)) S)))))))))))))) of role definition named breln1compE
% A new definition: (((eq Prop) breln1compE) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) (((breln1compset A) R) S))->((ex fofType) (fun (Xz:fofType)=> ((and ((and ((in Xz) A)) ((in ((kpair Xx) Xz)) R))) ((in ((kpair Xz) Xy)) S))))))))))))))
% Defined: breln1compE:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) (((breln1compset A) R) S))->((ex fofType) (fun (Xz:fofType)=> ((and ((and ((in Xz) A)) ((in ((kpair Xx) Xz)) R))) ((in ((kpair Xz) Xy)) S)))))))))))))
% FOF formula (<kernel.Constant object at 0xa06998>, <kernel.Sort object at 0xc80f38>) of role type named breln1compEex_type
% Using role type
% Declaring breln1compEex:Prop
% FOF formula (((eq Prop) breln1compEex) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) (((breln1compset A) R) S))->(forall (Xphi:Prop), ((forall (Xz:fofType), (((in Xz) A)->(((in ((kpair Xx) Xz)) R)->(((in ((kpair Xz) Xy)) S)->Xphi))))->Xphi)))))))))))) of role definition named breln1compEex
% A new definition: (((eq Prop) breln1compEex) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) (((breln1compset A) R) S))->(forall (Xphi:Prop), ((forall (Xz:fofType), (((in Xz) A)->(((in ((kpair Xx) Xz)) R)->(((in ((kpair Xz) Xy)) S)->Xphi))))->Xphi))))))))))))
% Defined: breln1compEex:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) (((breln1compset A) R) S))->(forall (Xphi:Prop), ((forall (Xz:fofType), (((in Xz) A)->(((in ((kpair Xx) Xz)) R)->(((in ((kpair Xz) Xy)) S)->Xphi))))->Xphi)))))))))))
% FOF formula (<kernel.Constant object at 0xa066c8>, <kernel.Sort object at 0xc80f38>) of role type named breln1unionprop_type
% Using role type
% Declaring breln1unionprop:Prop
% FOF formula (((eq Prop) breln1unionprop) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((breln1 A) ((binunion R) S))))))) of role definition named breln1unionprop
% A new definition: (((eq Prop) breln1unionprop) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((breln1 A) ((binunion R) S)))))))
% Defined: breln1unionprop:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((breln1 A) ((binunion R) S))))))
% FOF formula (<kernel.Constant object at 0xa06a70>, <kernel.Sort object at 0xc80f38>) of role type named breln1unionIL_type
% Using role type
% Declaring breln1unionIL:Prop
% FOF formula (((eq Prop) breln1unionIL) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))) of role definition named breln1unionIL
% A new definition: (((eq Prop) breln1unionIL) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))))
% Defined: breln1unionIL:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))
% FOF formula (<kernel.Constant object at 0xa06ab8>, <kernel.Sort object at 0xc80f38>) of role type named breln1unionIR_type
% Using role type
% Declaring breln1unionIR:Prop
% FOF formula (((eq Prop) breln1unionIR) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))) of role definition named breln1unionIR
% A new definition: (((eq Prop) breln1unionIR) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))))
% Defined: breln1unionIR:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))
% FOF formula (<kernel.Constant object at 0xa06e18>, <kernel.Sort object at 0xc80f38>) of role type named breln1unionI_type
% Using role type
% Declaring breln1unionI:Prop
% FOF formula (((eq Prop) breln1unionI) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))) of role definition named breln1unionI
% A new definition: (((eq Prop) breln1unionI) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))))
% Defined: breln1unionI:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))
% FOF formula (<kernel.Constant object at 0xa06950>, <kernel.Sort object at 0xc80f38>) of role type named breln1unionE_type
% Using role type
% Declaring breln1unionE:Prop
% FOF formula (((eq Prop) breln1unionE) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) ((binunion R) S))->((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))))))))))) of role definition named breln1unionE
% A new definition: (((eq Prop) breln1unionE) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) ((binunion R) S))->((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))))))))))))
% Defined: breln1unionE:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) ((binunion R) S))->((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))))))))))
% FOF formula (<kernel.Constant object at 0xa06518>, <kernel.Sort object at 0xc80f38>) of role type named breln1unionEcases_type
% Using role type
% Declaring breln1unionEcases:Prop
% FOF formula (((eq Prop) breln1unionEcases) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) ((binunion R) S))->(forall (Xphi:Prop), ((((in ((kpair Xx) Xy)) R)->Xphi)->((((in ((kpair Xx) Xy)) S)->Xphi)->Xphi))))))))))))) of role definition named breln1unionEcases
% A new definition: (((eq Prop) breln1unionEcases) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) ((binunion R) S))->(forall (Xphi:Prop), ((((in ((kpair Xx) Xy)) R)->Xphi)->((((in ((kpair Xx) Xy)) S)->Xphi)->Xphi)))))))))))))
% Defined: breln1unionEcases:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) ((binunion R) S))->(forall (Xphi:Prop), ((((in ((kpair Xx) Xy)) R)->Xphi)->((((in ((kpair Xx) Xy)) S)->Xphi)->Xphi))))))))))))
% FOF formula (<kernel.Constant object at 0xa060e0>, <kernel.Sort object at 0xc80f38>) of role type named breln1unionCommutes_type
% Using role type
% Declaring breln1unionCommutes:Prop
% FOF formula (((eq Prop) breln1unionCommutes) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(((eq fofType) ((binunion R) S)) ((binunion S) R))))))) of role definition named breln1unionCommutes
% A new definition: (((eq Prop) breln1unionCommutes) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(((eq fofType) ((binunion R) S)) ((binunion S) R)))))))
% Defined: breln1unionCommutes:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(((eq fofType) ((binunion R) S)) ((binunion S) R))))))
% FOF formula (<kernel.Constant object at 0xa065a8>, <kernel.Sort object at 0xc80f38>) of role type named woz2Ex_type
% Using role type
% Declaring woz2Ex:Prop
% FOF formula (((eq Prop) woz2Ex) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(((eq fofType) R) ((breln1invset A) ((breln1invset A) R)))))) of role definition named woz2Ex
% A new definition: (((eq Prop) woz2Ex) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(((eq fofType) R) ((breln1invset A) ((breln1invset A) R))))))
% Defined: woz2Ex:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(((eq fofType) R) ((breln1invset A) ((breln1invset A) R)))))
% FOF formula (<kernel.Constant object at 0xa06dd0>, <kernel.Sort object at 0xc80f38>) of role type named woz2W_type
% Using role type
% Declaring woz2W:Prop
% FOF formula (((eq Prop) woz2W) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(((eq fofType) ((breln1invset A) (((breln1compset A) R) S))) (((breln1compset A) ((breln1invset A) S)) ((breln1invset A) R)))))))) of role definition named woz2W
% A new definition: (((eq Prop) woz2W) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(((eq fofType) ((breln1invset A) (((breln1compset A) R) S))) (((breln1compset A) ((breln1invset A) S)) ((breln1invset A) R))))))))
% Defined: woz2W:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(((eq fofType) ((breln1invset A) (((breln1compset A) R) S))) (((breln1compset A) ((breln1invset A) S)) ((breln1invset A) R)))))))
% FOF formula (<kernel.Constant object at 0xa06488>, <kernel.Sort object at 0xc80f38>) of role type named woz2A_type
% Using role type
% Declaring woz2A:Prop
% FOF formula (((eq Prop) woz2A) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (T:fofType), (((breln1 A) T)->(((eq fofType) (((breln1compset A) ((binunion R) S)) T)) ((binunion (((breln1compset A) R) T)) (((breln1compset A) S) T)))))))))) of role definition named woz2A
% A new definition: (((eq Prop) woz2A) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (T:fofType), (((breln1 A) T)->(((eq fofType) (((breln1compset A) ((binunion R) S)) T)) ((binunion (((breln1compset A) R) T)) (((breln1compset A) S) T))))))))))
% Defined: woz2A:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (T:fofType), (((breln1 A) T)->(((eq fofType) (((breln1compset A) ((binunion R) S)) T)) ((binunion (((breln1compset A) R) T)) (((breln1compset A) S) T)))))))))
% FOF formula (<kernel.Constant object at 0xa06680>, <kernel.Sort object at 0xc80f38>) of role type named woz2B_type
% Using role type
% Declaring woz2B:Prop
% FOF formula (((eq Prop) woz2B) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (T:fofType), (((breln1 A) T)->(((eq fofType) (((breln1compset A) ((binunion R) S)) T)) ((binunion ((breln1invset A) (((breln1compset A) ((breln1invset A) T)) ((breln1invset A) S)))) ((breln1invset A) (((breln1compset A) ((breln1invset A) T)) ((breln1invset A) R)))))))))))) of role definition named woz2B
% A new definition: (((eq Prop) woz2B) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (T:fofType), (((breln1 A) T)->(((eq fofType) (((breln1compset A) ((binunion R) S)) T)) ((binunion ((breln1invset A) (((breln1compset A) ((breln1invset A) T)) ((breln1invset A) S)))) ((breln1invset A) (((breln1compset A) ((breln1invset A) T)) ((breln1invset A) R))))))))))))
% Defined: woz2B:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (T:fofType), (((breln1 A) T)->(((eq fofType) (((breln1compset A) ((binunion R) S)) T)) ((binunion ((breln1invset A) (((breln1compset A) ((breln1invset A) T)) ((breln1invset A) S)))) ((breln1invset A) (((breln1compset A) ((breln1invset A) T)) ((breln1invset A) R)))))))))))
% FOF formula (<kernel.Constant object at 0xa06908>, <kernel.Sort object at 0xc80f38>) of role type named image1Ex_type
% Using role type
% Declaring image1Ex:Prop
% FOF formula (((eq Prop) image1Ex) (forall (A:fofType) (Xf:(fofType->fofType)), ((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))))))) of role definition named image1Ex
% A new definition: (((eq Prop) image1Ex) (forall (A:fofType) (Xf:(fofType->fofType)), ((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))))))
% Defined: image1Ex:=(forall (A:fofType) (Xf:(fofType->fofType)), ((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))))))
% FOF formula (<kernel.Constant object at 0xa06d88>, <kernel.Sort object at 0xc80f38>) of role type named image1Ex1_type
% Using role type
% Declaring image1Ex1:Prop
% FOF formula (((eq Prop) image1Ex1) (forall (A:fofType) (Xf:(fofType->fofType)), (exu (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))))))) of role definition named image1Ex1
% A new definition: (((eq Prop) image1Ex1) (forall (A:fofType) (Xf:(fofType->fofType)), (exu (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))))))
% Defined: image1Ex1:=(forall (A:fofType) (Xf:(fofType->fofType)), (exu (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))))))
% FOF formula (<kernel.Constant object at 0xa06f80>, <kernel.DependentProduct object at 0xa06fc8>) of role type named image1_type
% Using role type
% Declaring image1:(fofType->((fofType->fofType)->fofType))
% FOF formula (<kernel.Constant object at 0xa067e8>, <kernel.Sort object at 0xc80f38>) of role type named image1Equiv_type
% Using role type
% Declaring image1Equiv:Prop
% FOF formula (((eq Prop) image1Equiv) (forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), ((iff ((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))) of role definition named image1Equiv
% A new definition: (((eq Prop) image1Equiv) (forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), ((iff ((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))))
% Defined: image1Equiv:=(forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), ((iff ((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))
% FOF formula (<kernel.Constant object at 0xa06f38>, <kernel.Sort object at 0xc80f38>) of role type named image1E_type
% Using role type
% Declaring image1E:Prop
% FOF formula (((eq Prop) image1E) (forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), (((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))) of role definition named image1E
% A new definition: (((eq Prop) image1E) (forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), (((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))))
% Defined: image1E:=(forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), (((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))
% FOF formula (<kernel.Constant object at 0xa06f80>, <kernel.Sort object at 0xc80f38>) of role type named image1I_type
% Using role type
% Declaring image1I:Prop
% FOF formula (((eq Prop) image1I) (forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))->((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))))) of role definition named image1I
% A new definition: (((eq Prop) image1I) (forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))->((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))))
% Defined: image1I:=(forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))->((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))))
% FOF formula (<kernel.Constant object at 0xa06bd8>, <kernel.DependentProduct object at 0xa080e0>) of role type named injective_type
% Using role type
% Declaring injective:(fofType->(fofType->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0xa06f80>, <kernel.DependentProduct object at 0xa080e0>) of role type named injFuncSet_type
% Using role type
% Declaring injFuncSet:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0xa06bd8>, <kernel.Sort object at 0xc80f38>) of role type named injFuncInInjFuncSet_type
% Using role type
% Declaring injFuncInInjFuncSet:Prop
% FOF formula (((eq Prop) injFuncInInjFuncSet) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->((((injective A) B) Xf)->((in Xf) ((injFuncSet A) B)))))) of role definition named injFuncInInjFuncSet
% A new definition: (((eq Prop) injFuncInInjFuncSet) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->((((injective A) B) Xf)->((in Xf) ((injFuncSet A) B))))))
% Defined: injFuncInInjFuncSet:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->((((injective A) B) Xf)->((in Xf) ((injFuncSet A) B)))))
% FOF formula (<kernel.Constant object at 0xa06bd8>, <kernel.Sort object at 0xc80f38>) of role type named injFuncSetFuncIn_type
% Using role type
% Declaring injFuncSetFuncIn:Prop
% FOF formula (((eq Prop) injFuncSetFuncIn) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((injFuncSet A) B))->((in Xf) ((funcSet A) B))))) of role definition named injFuncSetFuncIn
% A new definition: (((eq Prop) injFuncSetFuncIn) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((injFuncSet A) B))->((in Xf) ((funcSet A) B)))))
% Defined: injFuncSetFuncIn:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((injFuncSet A) B))->((in Xf) ((funcSet A) B))))
% FOF formula (<kernel.Constant object at 0xa08830>, <kernel.Sort object at 0xc80f38>) of role type named injFuncSetFuncInj_type
% Using role type
% Declaring injFuncSetFuncInj:Prop
% FOF formula (((eq Prop) injFuncSetFuncInj) (forall (Xx:fofType) (Xy:fofType) (Xf:fofType), (((in Xf) ((injFuncSet Xx) Xy))->(((injective Xx) Xy) Xf)))) of role definition named injFuncSetFuncInj
% A new definition: (((eq Prop) injFuncSetFuncInj) (forall (Xx:fofType) (Xy:fofType) (Xf:fofType), (((in Xf) ((injFuncSet Xx) Xy))->(((injective Xx) Xy) Xf))))
% Defined: injFuncSetFuncInj:=(forall (Xx:fofType) (Xy:fofType) (Xf:fofType), (((in Xf) ((injFuncSet Xx) Xy))->(((injective Xx) Xy) Xf)))
% FOF formula (<kernel.Constant object at 0xa083b0>, <kernel.DependentProduct object at 0xa08908>) of role type named surjective_type
% Using role type
% Declaring surjective:(fofType->(fofType->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0xa087a0>, <kernel.DependentProduct object at 0xa082d8>) of role type named surjFuncSet_type
% Using role type
% Declaring surjFuncSet:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0xa08830>, <kernel.Sort object at 0xc80f38>) of role type named surjFuncSetFuncIn_type
% Using role type
% Declaring surjFuncSetFuncIn:Prop
% FOF formula (((eq Prop) surjFuncSetFuncIn) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((surjFuncSet A) B))->((in Xf) ((funcSet A) B))))) of role definition named surjFuncSetFuncIn
% A new definition: (((eq Prop) surjFuncSetFuncIn) (forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((surjFuncSet A) B))->((in Xf) ((funcSet A) B)))))
% Defined: surjFuncSetFuncIn:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((surjFuncSet A) B))->((in Xf) ((funcSet A) B))))
% FOF formula (<kernel.Constant object at 0xa086c8>, <kernel.Sort object at 0xc80f38>) of role type named surjFuncSetFuncSurj_type
% Using role type
% Declaring surjFuncSetFuncSurj:Prop
% FOF formula (((eq Prop) surjFuncSetFuncSurj) (forall (Xx:fofType) (Xy:fofType) (Xf:fofType), (((in Xf) ((surjFuncSet Xx) Xy))->(((surjective Xx) Xy) Xf)))) of role definition named surjFuncSetFuncSurj
% A new definition: (((eq Prop) surjFuncSetFuncSurj) (forall (Xx:fofType) (Xy:fofType) (Xf:fofType), (((in Xf) ((surjFuncSet Xx) Xy))->(((surjective Xx) Xy) Xf))))
% Defined: surjFuncSetFuncSurj:=(forall (Xx:fofType) (Xy:fofType) (Xf:fofType), (((in Xf) ((surjFuncSet Xx) Xy))->(((surjective Xx) Xy) Xf)))
% FOF formula (<kernel.Constant object at 0xa08200>, <kernel.Sort object at 0xc80f38>) of role type named leftInvIsSurj_type
% Using role type
% Declaring leftInvIsSurj:Prop
% FOF formula (((eq Prop) leftInvIsSurj) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(forall (Xg:fofType), (((in Xg) ((funcSet B) A))->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap B) A) Xg) (Xf Xx))) Xx)))->(((surjective B) A) Xg))))))) of role definition named leftInvIsSurj
% A new definition: (((eq Prop) leftInvIsSurj) (forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(forall (Xg:fofType), (((in Xg) ((funcSet B) A))->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap B) A) Xg) (Xf Xx))) Xx)))->(((surjective B) A) Xg)))))))
% Defined: leftInvIsSurj:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(forall (Xg:fofType), (((in Xg) ((funcSet B) A))->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap B) A) Xg) (Xf Xx))) Xx)))->(((surjective B) A) Xg))))))
% FOF formula (<kernel.Constant object at 0xa087a0>, <kernel.Sort object at 0xc80f38>) of role type named surjCantorThm_type
% Using role type
% Declaring surjCantorThm:Prop
% FOF formula (((eq Prop) surjCantorThm) (forall (A:fofType) (Xf:fofType), (((in Xf) ((funcSet A) (powerset A)))->((((surjective A) (powerset A)) Xf)->False)))) of role definition named surjCantorThm
% A new definition: (((eq Prop) surjCantorThm) (forall (A:fofType) (Xf:fofType), (((in Xf) ((funcSet A) (powerset A)))->((((surjective A) (powerset A)) Xf)->False))))
% Defined: surjCantorThm:=(forall (A:fofType) (Xf:fofType), (((in Xf) ((funcSet A) (powerset A)))->((((surjective A) (powerset A)) Xf)->False)))
% FOF formula (<kernel.Constant object at 0xa08638>, <kernel.Sort object at 0xc80f38>) of role type named foundation2_type
% Using role type
% Declaring foundation2:Prop
% FOF formula (((eq Prop) foundation2) (forall (A:fofType), ((nonempty A)->((ex fofType) (fun (X:fofType)=> ((and ((in X) A)) (((eq fofType) ((binintersect X) A)) emptyset))))))) of role definition named foundation2
% A new definition: (((eq Prop) foundation2) (forall (A:fofType), ((nonempty A)->((ex fofType) (fun (X:fofType)=> ((and ((in X) A)) (((eq fofType) ((binintersect X) A)) emptyset)))))))
% Defined: foundation2:=(forall (A:fofType), ((nonempty A)->((ex fofType) (fun (X:fofType)=> ((and ((in X) A)) (((eq fofType) ((binintersect X) A)) emptyset))))))
% FOF formula (<kernel.Constant object at 0xa08518>, <kernel.Sort object at 0xc80f38>) of role type named notinself_type
% Using role type
% Declaring notinself:Prop
% FOF formula (((eq Prop) notinself) (forall (A:fofType), (((in A) A)->False))) of role definition named notinself
% A new definition: (((eq Prop) notinself) (forall (A:fofType), (((in A) A)->False)))
% Defined: notinself:=(forall (A:fofType), (((in A) A)->False))
% FOF formula (<kernel.Constant object at 0xa08098>, <kernel.Sort object at 0xc80f38>) of role type named notinself2_type
% Using role type
% Declaring notinself2:Prop
% FOF formula (((eq Prop) notinself2) (forall (A:fofType) (B:fofType), (((in A) B)->(((in B) A)->False)))) of role definition named notinself2
% A new definition: (((eq Prop) notinself2) (forall (A:fofType) (B:fofType), (((in A) B)->(((in B) A)->False))))
% Defined: notinself2:=(forall (A:fofType) (B:fofType), (((in A) B)->(((in B) A)->False)))
% FOF formula (<kernel.Constant object at 0xa08290>, <kernel.DependentProduct object at 0xa08758>) of role type named omegaS_type
% Using role type
% Declaring omegaS:(fofType->fofType)
% FOF formula (<kernel.Constant object at 0xa081b8>, <kernel.Sort object at 0xc80f38>) of role type named omegaSp_type
% Using role type
% Declaring omegaSp:Prop
% FOF formula (((eq Prop) omegaSp) (forall (Xx:fofType), (((in Xx) omega)->((in (omegaS Xx)) omega)))) of role definition named omegaSp
% A new definition: (((eq Prop) omegaSp) (forall (Xx:fofType), (((in Xx) omega)->((in (omegaS Xx)) omega))))
% Defined: omegaSp:=(forall (Xx:fofType), (((in Xx) omega)->((in (omegaS Xx)) omega)))
% FOF formula (<kernel.Constant object at 0xa08098>, <kernel.Sort object at 0xc80f38>) of role type named omegaSclos_type
% Using role type
% Declaring omegaSclos:Prop
% FOF formula (((eq Prop) omegaSclos) (forall (Xx:fofType), (((in Xx) omega)->((in ((setadjoin Xx) Xx)) omega)))) of role definition named omegaSclos
% A new definition: (((eq Prop) omegaSclos) (forall (Xx:fofType), (((in Xx) omega)->((in ((setadjoin Xx) Xx)) omega))))
% Defined: omegaSclos:=(forall (Xx:fofType), (((in Xx) omega)->((in ((setadjoin Xx) Xx)) omega)))
% FOF formula (<kernel.Constant object at 0xa08488>, <kernel.Sort object at 0xc80f38>) of role type named peano0notS_type
% Using role type
% Declaring peano0notS:Prop
% FOF formula (((eq Prop) peano0notS) (forall (Xx:fofType), (((in Xx) omega)->(not (((eq fofType) (omegaS Xx)) emptyset))))) of role definition named peano0notS
% A new definition: (((eq Prop) peano0notS) (forall (Xx:fofType), (((in Xx) omega)->(not (((eq fofType) (omegaS Xx)) emptyset)))))
% Defined: peano0notS:=(forall (Xx:fofType), (((in Xx) omega)->(not (((eq fofType) (omegaS Xx)) emptyset))))
% FOF formula (<kernel.Constant object at 0xa08b90>, <kernel.DependentProduct object at 0xa08bd8>) of role type named peano3_type
% Using role type
% Declaring peano3:(fofType->(fofType->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0xa083f8>, <kernel.DependentProduct object at 0xa08b48>) of role type named peano4_type
% Using role type
% Declaring peano4:(fofType->(fofType->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0xa08488>, <kernel.DependentProduct object at 0xa08b90>) of role type named peano5_type
% Using role type
% Declaring peano5:(fofType->(fofType->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0xa08368>, <kernel.Sort object at 0xc80f38>) of role type named peanoSinj_type
% Using role type
% Declaring peanoSinj:Prop
% FOF formula (((eq Prop) peanoSinj) (forall (Xx:fofType), (((in Xx) omega)->(forall (Xy:fofType), (((in Xy) omega)->((((eq fofType) (omegaS Xx)) (omegaS Xy))->(((eq fofType) Xx) Xy))))))) of role definition named peanoSinj
% A new definition: (((eq Prop) peanoSinj) (forall (Xx:fofType), (((in Xx) omega)->(forall (Xy:fofType), (((in Xy) omega)->((((eq fofType) (omegaS Xx)) (omegaS Xy))->(((eq fofType) Xx) Xy)))))))
% Defined: peanoSinj:=(forall (Xx:fofType), (((in Xx) omega)->(forall (Xy:fofType), (((in Xy) omega)->((((eq fofType) (omegaS Xx)) (omegaS Xy))->(((eq fofType) Xx) Xy))))))
% FOF formula (<kernel.Constant object at 0xa08a28>, <kernel.DependentProduct object at 0xa08e18>) of role type named peano_type
% Using role type
% Declaring peano:(fofType->(fofType->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0xa08170>, <kernel.DependentProduct object at 0xa083f8>) of role type named transitiveset_type
% Using role type
% Declaring transitiveset:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) transitiveset) (fun (A:fofType)=> (forall (X:fofType), (((in X) A)->((subset X) A))))) of role definition named transitiveset
% A new definition: (((eq (fofType->Prop)) transitiveset) (fun (A:fofType)=> (forall (X:fofType), (((in X) A)->((subset X) A)))))
% Defined: transitiveset:=(fun (A:fofType)=> (forall (X:fofType), (((in X) A)->((subset X) A))))
% FOF formula (<kernel.Constant object at 0xa083f8>, <kernel.Sort object at 0xc80f38>) of role type named transitivesetOp1_type
% Using role type
% Declaring transitivesetOp1:Prop
% FOF formula (((eq Prop) transitivesetOp1) (forall (X:fofType), ((transitiveset X)->(forall (A:fofType), (((in A) X)->((subset A) X)))))) of role definition named transitivesetOp1
% A new definition: (((eq Prop) transitivesetOp1) (forall (X:fofType), ((transitiveset X)->(forall (A:fofType), (((in A) X)->((subset A) X))))))
% Defined: transitivesetOp1:=(forall (X:fofType), ((transitiveset X)->(forall (A:fofType), (((in A) X)->((subset A) X)))))
% FOF formula (setextAx->(emptysetAx->(setadjoinAx->(powersetAx->(setunionAx->(omega0Ax->(omegaSAx->(omegaIndAx->(replAx->(foundationAx->(wellorderingAx->(descrp->(dsetconstrI->(dsetconstrEL->(dsetconstrER->(exuE1->(prop2setE->(emptysetE->(emptysetimpfalse->(notinemptyset->(exuE3e->(setext->(emptyI->(noeltsimpempty->(setbeta->(nonemptyE1->(nonemptyI->(nonemptyI1->(setadjoinIL->(emptyinunitempty->(setadjoinIR->(setadjoinE->(setadjoinOr->(setoftrueEq->(powersetI->(emptyinPowerset->(emptyInPowerset->(powersetE->(setunionI->(setunionE->(subPowSU->(exuE2->(nonemptyImpWitness->(uniqinunit->(notinsingleton->(eqinunit->(singletonsswitch->(upairsetE->(upairsetIL->(upairsetIR->(emptyE1->(vacuousDall->(quantDeMorgan1->(quantDeMorgan2->(quantDeMorgan3->(quantDeMorgan4->(prop2setI->(prop2set2propI->(notdexE->(notdallE->(exuI1->(exuI3->(exuI2->(inCongP->(in__Cong->(exuE3u->(exu__Cong->(emptyset__Cong->(setadjoin__Cong->(powerset__Cong->(setunion__Cong->(omega__Cong->(exuEu->(descr__Cong->(dsetconstr__Cong->(subsetI1->(eqimpsubset2->(eqimpsubset1->(subsetI2->(emptysetsubset->(subsetE->(subsetE2->(notsubsetI->(notequalI1->(notequalI2->(subsetRefl->(subsetTrans->(setadjoinSub->(setadjoinSub2->(subset2powerset->(setextsub->(subsetemptysetimpeq->(powersetI1->(powersetE1->(inPowerset->(powersetsubset->(sepInPowerset->(sepSubset->(binunionIL->(upairset2IR->(binunionIR->(binunionEcases->(binunionE->(binunionLsub->(binunionRsub->(binintersectI->(binintersectSubset5->(binintersectEL->(binintersectLsub->(binintersectSubset2->(binintersectSubset3->(binintersectER->(disjointsetsI1->(binintersectRsub->(binintersectSubset4->(binintersectSubset1->(bs114d->(setminusI->(setminusEL->(setminusER->(setminusSubset2->(setminusERneg->(setminusELneg->(setminusILneg->(setminusIRneg->(setminusLsub->(setminusSubset1->(symdiffE->(symdiffI1->(symdiffI2->(symdiffIneg1->(symdiffIneg2->(secondinupair->(setukpairIL->(setukpairIR->(kpairiskpair->(kpairp->(singletonsubset->(singletoninpowerset->(singletoninpowunion->(upairset2E->(upairsubunion->(upairinpowunion->(ubforcartprodlem1->(ubforcartprodlem2->(ubforcartprodlem3->(cartprodpairin->(cartprodmempair1->(cartprodmempair->(setunionE2->(setunionsingleton1->(setunionsingleton2->(setunionsingleton->(singletonprop->(ex1E1->(ex1I->(ex1I2->(singletonsuniq->(setukpairinjL1->(kfstsingleton->(theprop->(kfstpairEq->(cartprodfstin->(setukpairinjL2->(setukpairinjL->(setukpairinjR11->(setukpairinjR12->(setukpairinjR1->(upairequniteq->(setukpairinjR2->(setukpairinjR->(ksndsingleton->(ksndpairEq->(kpairsurjEq->(cartprodsndin->(cartprodpairmemEL->(cartprodpairmemER->(cartprodmempaircEq->(cartprodfstpairEq->(cartprodsndpairEq->(cartprodpairsurjEq->(dpsetconstrI->(dpsetconstrSub->(setOfPairsIsBReln->(dpsetconstrERa->(dpsetconstrEL1->(dpsetconstrEL2->(dpsetconstrER->(funcImageSingleton->(apProp->(app->(infuncsetfunc->(ap2p->(funcinfuncset->(lamProp->(lamp->(lam2p->(brelnall1->(brelnall2->(ex1E2->(funcGraphProp1->(funcGraphProp3->(funcGraphProp2->(funcextLem->(funcGraphProp4->(subbreln->(eqbreln->(funcext->(funcext2->(ap2apEq1->(ap2apEq2->(beta1->(eta1->(lam2lamEq->(beta2->(eta2->(iffalseProp1->(iffalseProp2->(iftrueProp1->(iftrueProp2->(ifSingleton->(ifp->(theeq->(iftrue->(iffalse->(iftrueorfalse->(binintersectT_lem->(binunionT_lem->(powersetT_lem->(setminusT_lem->(complementT_lem->(setextT->(subsetTI->(powersetTI1->(powersetTE1->(complementTI1->(complementTE1->(binintersectTELcontra->(binintersectTERcontra->(contrasubsetT->(contrasubsetT1->(contrasubsetT2->(contrasubsetT3->(doubleComplementI1->(doubleComplementE1->(doubleComplementSub1->(doubleComplementSub2->(doubleComplementEq->(complementTnotintersectT->(complementImpComplementIntersect->(complementSubsetComplementIntersect->(complementInPowersetComplementIntersect->(contraSubsetComplement->(complementTcontraSubset->(binunionTILcontra->(binunionTIRcontra->(inIntersectImpInUnion->(inIntersectImpInUnion2->(inIntersectImpInIntersectUnions->(intersectInPowersetIntersectUnions->(inComplementUnionImpNotIn1->(inComplementUnionImpInComplement1->(binunionTE->(binunionTEcontra->(demorgan2a1->(complementUnionInPowersetComplement->(demorgan2a2->(demorgan1a->(demorgan1b->(demorgan1->(demorgan2a->(demorgan2b2->(demorgan2b->(demorgan2->(woz13rule0->(woz13rule1->(woz13rule2->(woz13rule3->(woz13rule4->(woz1_1->(woz1_2->(woz1_3->(woz1_4->(woz1_5->(breln1all2->(breln1SetBreln1->(choice2fnsingleton->(setOfPairsIsBReln1->(breln1all1->(subbreln1->(eqbreln1->(breln1invprop->(breln1invI->(breln1invE->(breln1compprop->(breln1compI->(breln1compE->(breln1compEex->(breln1unionprop->(breln1unionIL->(breln1unionIR->(breln1unionI->(breln1unionE->(breln1unionEcases->(breln1unionCommutes->(woz2Ex->(woz2W->(woz2A->(woz2B->(image1Ex->(image1Ex1->(image1Equiv->(image1E->(image1I->(injFuncInInjFuncSet->(injFuncSetFuncIn->(injFuncSetFuncInj->(surjFuncSetFuncIn->(surjFuncSetFuncSurj->(leftInvIsSurj->(surjCantorThm->(foundation2->(notinself->(notinself2->(omegaSp->(omegaSclos->(peano0notS->(peanoSinj->(transitivesetOp1->(forall (X:fofType), ((transitiveset X)->(forall (Y:fofType), ((transitiveset Y)->(transitiveset ((binintersect X) Y))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) of role conjecture named binintTransitive
% Conjecture to prove = (setextAx->(emptysetAx->(setadjoinAx->(powersetAx->(setunionAx->(omega0Ax->(omegaSAx->(omegaIndAx->(replAx->(foundationAx->(wellorderingAx->(descrp->(dsetconstrI->(dsetconstrEL->(dsetconstrER->(exuE1->(prop2setE->(emptysetE->(emptysetimpfalse->(notinemptyset->(exuE3e->(setext->(emptyI->(noeltsimpempty->(setbeta->(nonemptyE1->(nonemptyI->(nonemptyI1->(setadjoinIL->(emptyinunitempty->(setadjoinIR->(setadjoinE->(setadjoinOr->(setoftrueEq->(powersetI->(emptyinPowerset->(emptyInPowerset->(powersetE->(setunionI->(setunionE->(subPowSU->(exuE2->(nonemptyImpWitness->(uniqinunit->(notinsingleton->(eqinunit->(singletonsswitch->(upairsetE->(upairsetIL->(upairsetIR->(emptyE1->(vacuousDall->(quantDeMorgan1->(quantDeMorgan2->(quantDeMorgan3->(quantDeMorgan4->(prop2setI->(prop2set2propI->(notdexE->(notdallE->(exuI1->(exuI3->(exuI2->(inCongP->(in__Cong->(exuE3u->(exu__Cong->(emptyset__Cong->(setadjoin__Cong->(powerset__Cong->(setunion__Cong->(omega__Cong->(exuEu->(descr__Cong->(dsetconstr__Cong->(subsetI1->(eqimpsubset2->(eqimpsubset1->(subsetI2->(emptysetsubset->(subsetE->(subsetE2->(notsubsetI->(notequalI1->(notequalI2->(subsetRefl->(subsetTrans->(setadjoinSub->(setadjoinSub2->(subset2powerset->(setextsub->(subsetemptysetimpeq->(powersetI1->(powersetE1->(inPowerset->(powersetsubset->(sepInPowerset->(sepSubset->(binunionIL->(upairset2IR->(binunionIR->(binunionEcases->(binunionE->(binunionLsub->(binunionRsub->(binintersectI->(binintersectSubset5->(binintersectEL->(binintersectLsub->(binintersectSubset2->(binintersectSubset3->(binintersectER->(disjointsetsI1->(binintersectRsub->(binintersectSubset4->(binintersectSubset1->(bs114d->(setminusI->(setminusEL->(setminusER->(setminusSubset2->(setminusERneg->(setminusELneg->(setminusILneg->(setminusIRneg->(setminusLsub->(setminusSubset1->(symdiffE->(symdiffI1->(symdiffI2->(symdiffIneg1->(symdiffIneg2->(secondinupair->(setukpairIL->(setukpairIR->(kpairiskpair->(kpairp->(singletonsubset->(singletoninpowerset->(singletoninpowunion->(upairset2E->(upairsubunion->(upairinpowunion->(ubforcartprodlem1->(ubforcartprodlem2->(ubforcartprodlem3->(cartprodpairin->(cartprodmempair1->(cartprodmempair->(setunionE2->(setunionsingleton1->(setunionsingleton2->(setunionsingleton->(singletonprop->(ex1E1->(ex1I->(ex1I2->(singletonsuniq->(setukpairinjL1->(kfstsingleton->(theprop->(kfstpairEq->(cartprodfstin->(setukpairinjL2->(setukpairinjL->(setukpairinjR11->(setukpairinjR12->(setukpairinjR1->(upairequniteq->(setukpairinjR2->(setukpairinjR->(ksndsingleton->(ksndpairEq->(kpairsurjEq->(cartprodsndin->(cartprodpairmemEL->(cartprodpairmemER->(cartprodmempaircEq->(cartprodfstpairEq->(cartprodsndpairEq->(cartprodpairsurjEq->(dpsetconstrI->(dpsetconstrSub->(setOfPairsIsBReln->(dpsetconstrERa->(dpsetconstrEL1->(dpsetconstrEL2->(dpsetconstrER->(funcImageSingleton->(apProp->(app->(infuncsetfunc->(ap2p->(funcinfuncset->(lamProp->(lamp->(lam2p->(brelnall1->(brelnall2->(ex1E2->(funcGraphProp1->(funcGraphProp3->(funcGraphProp2->(funcextLem->(funcGraphProp4->(subbreln->(eqbreln->(funcext->(funcext2->(ap2apEq1->(ap2apEq2->(beta1->(eta1->(lam2lamEq->(beta2->(eta2->(iffalseProp1->(iffalseProp2->(iftrueProp1->(iftrueProp2->(ifSingleton->(ifp->(theeq->(iftrue->(iffalse->(iftrueorfalse->(binintersectT_lem->(binunionT_lem->(powersetT_lem->(setminusT_lem->(complementT_lem->(setextT->(subsetTI->(powersetTI1->(powersetTE1->(complementTI1->(complementTE1->(binintersectTELcontra->(binintersectTERcontra->(contrasubsetT->(contrasubsetT1->(contrasubsetT2->(contrasubsetT3->(doubleComplementI1->(doubleComplementE1->(doubleComplementSub1->(doubleComplementSub2->(doubleComplementEq->(complementTnotintersectT->(complementImpComplementIntersect->(complementSubsetComplementIntersect->(complementInPowersetComplementIntersect->(contraSubsetComplement->(complementTcontraSubset->(binunionTILcontra->(binunionTIRcontra->(inIntersectImpInUnion->(inIntersectImpInUnion2->(inIntersectImpInIntersectUnions->(intersectInPowersetIntersectUnions->(inComplementUnionImpNotIn1->(inComplementUnionImpInComplement1->(binunionTE->(binunionTEcontra->(demorgan2a1->(complementUnionInPowersetComplement->(demorgan2a2->(demorgan1a->(demorgan1b->(demorgan1->(demorgan2a->(demorgan2b2->(demorgan2b->(demorgan2->(woz13rule0->(woz13rule1->(woz13rule2->(woz13rule3->(woz13rule4->(woz1_1->(woz1_2->(woz1_3->(woz1_4->(woz1_5->(breln1all2->(breln1SetBreln1->(choice2fnsingleton->(setOfPairsIsBReln1->(breln1all1->(subbreln1->(eqbreln1->(breln1invprop->(breln1invI->(breln1invE->(breln1compprop->(breln1compI->(breln1compE->(breln1compEex->(breln1unionprop->(breln1unionIL->(breln1unionIR->(breln1unionI->(breln1unionE->(breln1unionEcases->(breln1unionCommutes->(woz2Ex->(woz2W->(woz2A->(woz2B->(image1Ex->(image1Ex1->(image1Equiv->(image1E->(image1I->(injFuncInInjFuncSet->(injFuncSetFuncIn->(injFuncSetFuncInj->(surjFuncSetFuncIn->(surjFuncSetFuncSurj->(leftInvIsSurj->(surjCantorThm->(foundation2->(notinself->(notinself2->(omegaSp->(omegaSclos->(peano0notS->(peanoSinj->(transitivesetOp1->(forall (X:fofType), ((transitiveset X)->(forall (Y:fofType), ((transitiveset Y)->(transitiveset ((binintersect X) Y))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))):Prop
% We need to prove ['(setextAx->(emptysetAx->(setadjoinAx->(powersetAx->(setunionAx->(omega0Ax->(omegaSAx->(omegaIndAx->(replAx->(foundationAx->(wellorderingAx->(descrp->(dsetconstrI->(dsetconstrEL->(dsetconstrER->(exuE1->(prop2setE->(emptysetE->(emptysetimpfalse->(notinemptyset->(exuE3e->(setext->(emptyI->(noeltsimpempty->(setbeta->(nonemptyE1->(nonemptyI->(nonemptyI1->(setadjoinIL->(emptyinunitempty->(setadjoinIR->(setadjoinE->(setadjoinOr->(setoftrueEq->(powersetI->(emptyinPowerset->(emptyInPowerset->(powersetE->(setunionI->(setunionE->(subPowSU->(exuE2->(nonemptyImpWitness->(uniqinunit->(notinsingleton->(eqinunit->(singletonsswitch->(upairsetE->(upairsetIL->(upairsetIR->(emptyE1->(vacuousDall->(quantDeMorgan1->(quantDeMorgan2->(quantDeMorgan3->(quantDeMorgan4->(prop2setI->(prop2set2propI->(notdexE->(notdallE->(exuI1->(exuI3->(exuI2->(inCongP->(in__Cong->(exuE3u->(exu__Cong->(emptyset__Cong->(setadjoin__Cong->(powerset__Cong->(setunion__Cong->(omega__Cong->(exuEu->(descr__Cong->(dsetconstr__Cong->(subsetI1->(eqimpsubset2->(eqimpsubset1->(subsetI2->(emptysetsubset->(subsetE->(subsetE2->(notsubsetI->(notequalI1->(notequalI2->(subsetRefl->(subsetTrans->(setadjoinSub->(setadjoinSub2->(subset2powerset->(setextsub->(subsetemptysetimpeq->(powersetI1->(powersetE1->(inPowerset->(powersetsubset->(sepInPowerset->(sepSubset->(binunionIL->(upairset2IR->(binunionIR->(binunionEcases->(binunionE->(binunionLsub->(binunionRsub->(binintersectI->(binintersectSubset5->(binintersectEL->(binintersectLsub->(binintersectSubset2->(binintersectSubset3->(binintersectER->(disjointsetsI1->(binintersectRsub->(binintersectSubset4->(binintersectSubset1->(bs114d->(setminusI->(setminusEL->(setminusER->(setminusSubset2->(setminusERneg->(setminusELneg->(setminusILneg->(setminusIRneg->(setminusLsub->(setminusSubset1->(symdiffE->(symdiffI1->(symdiffI2->(symdiffIneg1->(symdiffIneg2->(secondinupair->(setukpairIL->(setukpairIR->(kpairiskpair->(kpairp->(singletonsubset->(singletoninpowerset->(singletoninpowunion->(upairset2E->(upairsubunion->(upairinpowunion->(ubforcartprodlem1->(ubforcartprodlem2->(ubforcartprodlem3->(cartprodpairin->(cartprodmempair1->(cartprodmempair->(setunionE2->(setunionsingleton1->(setunionsingleton2->(setunionsingleton->(singletonprop->(ex1E1->(ex1I->(ex1I2->(singletonsuniq->(setukpairinjL1->(kfstsingleton->(theprop->(kfstpairEq->(cartprodfstin->(setukpairinjL2->(setukpairinjL->(setukpairinjR11->(setukpairinjR12->(setukpairinjR1->(upairequniteq->(setukpairinjR2->(setukpairinjR->(ksndsingleton->(ksndpairEq->(kpairsurjEq->(cartprodsndin->(cartprodpairmemEL->(cartprodpairmemER->(cartprodmempaircEq->(cartprodfstpairEq->(cartprodsndpairEq->(cartprodpairsurjEq->(dpsetconstrI->(dpsetconstrSub->(setOfPairsIsBReln->(dpsetconstrERa->(dpsetconstrEL1->(dpsetconstrEL2->(dpsetconstrER->(funcImageSingleton->(apProp->(app->(infuncsetfunc->(ap2p->(funcinfuncset->(lamProp->(lamp->(lam2p->(brelnall1->(brelnall2->(ex1E2->(funcGraphProp1->(funcGraphProp3->(funcGraphProp2->(funcextLem->(funcGraphProp4->(subbreln->(eqbreln->(funcext->(funcext2->(ap2apEq1->(ap2apEq2->(beta1->(eta1->(lam2lamEq->(beta2->(eta2->(iffalseProp1->(iffalseProp2->(iftrueProp1->(iftrueProp2->(ifSingleton->(ifp->(theeq->(iftrue->(iffalse->(iftrueorfalse->(binintersectT_lem->(binunionT_lem->(powersetT_lem->(setminusT_lem->(complementT_lem->(setextT->(subsetTI->(powersetTI1->(powersetTE1->(complementTI1->(complementTE1->(binintersectTELcontra->(binintersectTERcontra->(contrasubsetT->(contrasubsetT1->(contrasubsetT2->(contrasubsetT3->(doubleComplementI1->(doubleComplementE1->(doubleComplementSub1->(doubleComplementSub2->(doubleComplementEq->(complementTnotintersectT->(complementImpComplementIntersect->(complementSubsetComplementIntersect->(complementInPowersetComplementIntersect->(contraSubsetComplement->(complementTcontraSubset->(binunionTILcontra->(binunionTIRcontra->(inIntersectImpInUnion->(inIntersectImpInUnion2->(inIntersectImpInIntersectUnions->(intersectInPowersetIntersectUnions->(inComplementUnionImpNotIn1->(inComplementUnionImpInComplement1->(binunionTE->(binunionTEcontra->(demorgan2a1->(complementUnionInPowersetComplement->(demorgan2a2->(demorgan1a->(demorgan1b->(demorgan1->(demorgan2a->(demorgan2b2->(demorgan2b->(demorgan2->(woz13rule0->(woz13rule1->(woz13rule2->(woz13rule3->(woz13rule4->(woz1_1->(woz1_2->(woz1_3->(woz1_4->(woz1_5->(breln1all2->(breln1SetBreln1->(choice2fnsingleton->(setOfPairsIsBReln1->(breln1all1->(subbreln1->(eqbreln1->(breln1invprop->(breln1invI->(breln1invE->(breln1compprop->(breln1compI->(breln1compE->(breln1compEex->(breln1unionprop->(breln1unionIL->(breln1unionIR->(breln1unionI->(breln1unionE->(breln1unionEcases->(breln1unionCommutes->(woz2Ex->(woz2W->(woz2A->(woz2B->(image1Ex->(image1Ex1->(image1Equiv->(image1E->(image1I->(injFuncInInjFuncSet->(injFuncSetFuncIn->(injFuncSetFuncInj->(surjFuncSetFuncIn->(surjFuncSetFuncSurj->(leftInvIsSurj->(surjCantorThm->(foundation2->(notinself->(notinself2->(omegaSp->(omegaSclos->(peano0notS->(peanoSinj->(transitivesetOp1->(forall (X:fofType), ((transitiveset X)->(forall (Y:fofType), ((transitiveset Y)->(transitiveset ((binintersect X) Y)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))']
% Parameter fofType:Type.
% Parameter in:(fofType->(fofType->Prop)).
% Definition exu:=(fun (Xphi:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))):((fofType->Prop)->Prop).
% Definition setextAx:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), ((iff ((in Xx) A)) ((in Xx) B)))->(((eq fofType) A) B))):Prop.
% Parameter emptyset:fofType.
% Definition emptysetAx:=(forall (Xx:fofType), (((in Xx) emptyset)->False)):Prop.
% Parameter setadjoin:(fofType->(fofType->fofType)).
% Definition setadjoinAx:=(forall (Xx:fofType) (A:fofType) (Xy:fofType), ((iff ((in Xy) ((setadjoin Xx) A))) ((or (((eq fofType) Xy) Xx)) ((in Xy) A)))):Prop.
% Parameter powerset:(fofType->fofType).
% Definition powersetAx:=(forall (A:fofType) (B:fofType), ((iff ((in B) (powerset A))) (forall (Xx:fofType), (((in Xx) B)->((in Xx) A))))):Prop.
% Parameter setunion:(fofType->fofType).
% Definition setunionAx:=(forall (A:fofType) (Xx:fofType), ((iff ((in Xx) (setunion A))) ((ex fofType) (fun (B:fofType)=> ((and ((in Xx) B)) ((in B) A)))))):Prop.
% Parameter omega:fofType.
% Definition omega0Ax:=((in emptyset) omega):Prop.
% Definition omegaSAx:=(forall (Xx:fofType), (((in Xx) omega)->((in ((setadjoin Xx) Xx)) omega))):Prop.
% Definition omegaIndAx:=(forall (A:fofType), (((and ((in emptyset) A)) (forall (Xx:fofType), (((and ((in Xx) omega)) ((in Xx) A))->((in ((setadjoin Xx) Xx)) A))))->(forall (Xx:fofType), (((in Xx) omega)->((in Xx) A))))):Prop.
% Definition replAx:=(forall (Xphi:(fofType->(fofType->Prop))) (A:fofType), ((forall (Xx:fofType), (((in Xx) A)->(exu (fun (Xy:fofType)=> ((Xphi Xx) Xy)))))->((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) ((Xphi Xy) Xx)))))))))):Prop.
% Definition foundationAx:=(forall (A:fofType), (((ex fofType) (fun (Xx:fofType)=> ((in Xx) A)))->((ex fofType) (fun (B:fofType)=> ((and ((in B) A)) (((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) B)) ((in Xx) A))))->False)))))):Prop.
% Definition wellorderingAx:=(forall (A:fofType), ((ex fofType) (fun (B:fofType)=> ((and ((and ((and (forall (C:fofType), (((in C) B)->(forall (Xx:fofType), (((in Xx) C)->((in Xx) A)))))) (forall (Xx:fofType) (Xy:fofType), (((and ((in Xx) A)) ((in Xy) A))->((forall (C:fofType), (((in C) B)->((iff ((in Xx) C)) ((in Xy) C))))->(((eq fofType) Xx) Xy)))))) (forall (C:fofType) (D:fofType), (((and ((in C) B)) ((in D) B))->((or (forall (Xx:fofType), (((in Xx) C)->((in Xx) D)))) (forall (Xx:fofType), (((in Xx) D)->((in Xx) C)))))))) (forall (C:fofType), (((and (forall (Xx:fofType), (((in Xx) C)->((in Xx) A)))) ((ex fofType) (fun (Xx:fofType)=> ((in Xx) C))))->((ex fofType) (fun (D:fofType)=> ((ex fofType) (fun (Xx:fofType)=> ((and ((and ((and ((in D) B)) ((in Xx) C))) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) D)) ((in Xy) C))))->False))) (forall (E:fofType), (((in E) B)->((or (forall (Xy:fofType), (((in Xy) E)->((in Xy) D)))) ((in Xx) E))))))))))))))):Prop.
% Parameter descr:((fofType->Prop)->fofType).
% Definition descrp:=(forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->(Xphi (descr (fun (Xx:fofType)=> (Xphi Xx)))))):Prop.
% Parameter dsetconstr:(fofType->((fofType->Prop)->fofType)).
% Definition dsetconstrI:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))))):Prop.
% Definition dsetconstrEL:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))->((in Xx) A))):Prop.
% Definition dsetconstrER:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))->(Xphi Xx))):Prop.
% Definition exuE1:=(forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))))):Prop.
% Parameter prop2set:(Prop->fofType).
% Definition prop2setE:=(forall (Xphi:Prop) (Xx:fofType), (((in Xx) (prop2set Xphi))->Xphi)):Prop.
% Definition emptysetE:=(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Xphi:Prop), Xphi))):Prop.
% Definition emptysetimpfalse:=(forall (Xx:fofType), (((in Xx) emptyset)->False)):Prop.
% Definition notinemptyset:=(forall (Xx:fofType), (((in Xx) emptyset)->False)):Prop.
% Definition exuE3e:=(forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> (Xphi Xx))))):Prop.
% Definition setext:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->(((eq fofType) A) B)))):Prop.
% Definition emptyI:=(forall (A:fofType), ((forall (Xx:fofType), (((in Xx) A)->False))->(((eq fofType) A) emptyset))):Prop.
% Definition noeltsimpempty:=(forall (A:fofType), ((forall (Xx:fofType), (((in Xx) A)->False))->(((eq fofType) A) emptyset))):Prop.
% Definition setbeta:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((iff ((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))) (Xphi Xx)))):Prop.
% Definition nonempty:=(fun (Xx:fofType)=> (not (((eq fofType) Xx) emptyset))):(fofType->Prop).
% Definition nonemptyE1:=(forall (A:fofType), ((nonempty A)->((ex fofType) (fun (Xx:fofType)=> ((in Xx) A))))):Prop.
% Definition nonemptyI:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->(nonempty ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))))):Prop.
% Definition nonemptyI1:=(forall (A:fofType), (((ex fofType) (fun (Xx:fofType)=> ((in Xx) A)))->(nonempty A))):Prop.
% Definition setadjoinIL:=(forall (Xx:fofType) (Xy:fofType), ((in Xx) ((setadjoin Xx) Xy))):Prop.
% Definition emptyinunitempty:=((in emptyset) ((setadjoin emptyset) emptyset)):Prop.
% Definition setadjoinIR:=(forall (Xx:fofType) (A:fofType) (Xy:fofType), (((in Xy) A)->((in Xy) ((setadjoin Xx) A)))):Prop.
% Definition setadjoinE:=(forall (Xx:fofType) (A:fofType) (Xy:fofType), (((in Xy) ((setadjoin Xx) A))->(forall (Xphi:Prop), (((((eq fofType) Xy) Xx)->Xphi)->((((in Xy) A)->Xphi)->Xphi))))):Prop.
% Definition setadjoinOr:=(forall (Xx:fofType) (A:fofType) (Xy:fofType), (((in Xy) ((setadjoin Xx) A))->((or (((eq fofType) Xy) Xx)) ((in Xy) A)))):Prop.
% Definition setoftrueEq:=(forall (A:fofType), (((eq fofType) ((dsetconstr A) (fun (Xx:fofType)=> True))) A)):Prop.
% Definition powersetI:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) B)->((in Xx) A)))->((in B) (powerset A)))):Prop.
% Definition emptyinPowerset:=(forall (A:fofType), ((in emptyset) (powerset A))):Prop.
% Definition emptyInPowerset:=(forall (A:fofType), ((in emptyset) (powerset A))):Prop.
% Definition powersetE:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in B) (powerset A))->(((in Xx) B)->((in Xx) A)))):Prop.
% Definition setunionI:=(forall (A:fofType) (Xx:fofType) (B:fofType), (((in Xx) B)->(((in B) A)->((in Xx) (setunion A))))):Prop.
% Definition setunionE:=(forall (A:fofType) (Xx:fofType), (((in Xx) (setunion A))->(forall (Xphi:Prop), ((forall (B:fofType), (((in Xx) B)->(((in B) A)->Xphi)))->Xphi)))):Prop.
% Definition subPowSU:=(forall (A:fofType) (Xx:fofType), (((in Xx) A)->((in Xx) (powerset (setunion A))))):Prop.
% Definition exuE2:=(forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx))))))):Prop.
% Definition nonemptyImpWitness:=(forall (A:fofType), ((nonempty A)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) True))))):Prop.
% Definition uniqinunit:=(forall (Xx:fofType) (Xy:fofType), (((in Xx) ((setadjoin Xy) emptyset))->(((eq fofType) Xx) Xy))):Prop.
% Definition notinsingleton:=(forall (Xx:fofType) (Xy:fofType), ((not (((eq fofType) Xx) Xy))->(((in Xy) ((setadjoin Xx) emptyset))->False))):Prop.
% Definition eqinunit:=(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((in Xx) ((setadjoin Xy) emptyset)))):Prop.
% Definition singletonsswitch:=(forall (Xx:fofType) (Xy:fofType), (((in Xx) ((setadjoin Xy) emptyset))->((in Xy) ((setadjoin Xx) emptyset)))):Prop.
% Definition upairsetE:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((in Xz) ((setadjoin Xx) ((setadjoin Xy) emptyset)))->((or (((eq fofType) Xz) Xx)) (((eq fofType) Xz) Xy)))):Prop.
% Definition upairsetIL:=(forall (Xx:fofType) (Xy:fofType), ((in Xx) ((setadjoin Xx) ((setadjoin Xy) emptyset)))):Prop.
% Definition upairsetIR:=(forall (Xx:fofType) (Xy:fofType), ((in Xy) ((setadjoin Xx) ((setadjoin Xy) emptyset)))):Prop.
% Definition emptyE1:=(forall (A:fofType) (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->((((eq fofType) ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) emptyset)->False))):Prop.
% Definition vacuousDall:=(forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx))):Prop.
% Definition quantDeMorgan1:=(forall (A:fofType) (Xphi:(fofType->Prop)), (((forall (Xx:fofType), (((in Xx) A)->(Xphi Xx)))->False)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((Xphi Xx)->False)))))):Prop.
% Definition quantDeMorgan2:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->((Xphi Xx)->False)))->(((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->False))):Prop.
% Definition quantDeMorgan3:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->False)->(forall (Xx:fofType), (((in Xx) A)->((Xphi Xx)->False))))):Prop.
% Definition quantDeMorgan4:=(forall (A:fofType) (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((Xphi Xx)->False))))->((forall (Xx:fofType), (((in Xx) A)->(Xphi Xx)))->False))):Prop.
% Definition prop2setI:=(forall (Xphi:Prop), (Xphi->((in emptyset) (prop2set Xphi)))):Prop.
% Parameter set2prop:(fofType->Prop).
% Definition prop2set2propI:=(forall (Xphi:Prop), (Xphi->(set2prop (prop2set Xphi)))):Prop.
% Definition notdexE:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->False)->(forall (Xx:fofType), (((in Xx) A)->((Xphi Xx)->False))))):Prop.
% Definition notdallE:=(forall (A:fofType) (Xphi:(fofType->Prop)), (((forall (Xx:fofType), (((in Xx) A)->(Xphi Xx)))->False)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((Xphi Xx)->False)))))):Prop.
% Definition exuI1:=(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))->(exu (fun (Xx:fofType)=> (Xphi Xx))))):Prop.
% Definition exuI3:=(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (Xphi Xx)))->((forall (Xx:fofType) (Xy:fofType), ((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy))))->(exu (fun (Xx:fofType)=> (Xphi Xx)))))):Prop.
% Definition exuI2:=(forall (Xphi:(fofType->Prop)), (((ex fofType) (fun (Xx:fofType)=> (forall (Xy:fofType), ((iff (Xphi Xy)) (((eq fofType) Xy) Xx)))))->(exu (fun (Xx:fofType)=> (Xphi Xx))))):Prop.
% Definition inCongP:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(((in Xx) A)->((in Xy) B)))))):Prop.
% Definition in__Cong:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((iff ((in Xx) A)) ((in Xy) B)))))):Prop.
% Definition exuE3u:=(forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->(forall (Xx:fofType) (Xy:fofType), ((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))):Prop.
% Definition exu__Cong:=(forall (Xphi:(fofType->Prop)) (Xpsi:(fofType->Prop)), ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((iff (Xphi Xx)) (Xpsi Xy))))->((iff (exu (fun (Xx:fofType)=> (Xphi Xx)))) (exu (fun (Xx:fofType)=> (Xpsi Xx)))))):Prop.
% Definition emptyset__Cong:=(((eq fofType) emptyset) emptyset):Prop.
% Definition setadjoin__Cong:=(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(forall (Xz:fofType) (Xu:fofType), ((((eq fofType) Xz) Xu)->(((eq fofType) ((setadjoin Xx) Xz)) ((setadjoin Xy) Xu)))))):Prop.
% Definition powerset__Cong:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(((eq fofType) (powerset A)) (powerset B)))):Prop.
% Definition setunion__Cong:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(((eq fofType) (setunion A)) (setunion B)))):Prop.
% Definition omega__Cong:=(((eq fofType) omega) omega):Prop.
% Definition exuEu:=(forall (Xphi:(fofType->Prop)), ((exu (fun (Xx:fofType)=> (Xphi Xx)))->(forall (Xx:fofType) (Xy:fofType), ((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))):Prop.
% Definition descr__Cong:=(forall (Xphi:(fofType->Prop)) (Xpsi:(fofType->Prop)), ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->((iff (Xphi Xx)) (Xpsi Xy))))->((exu (fun (Xx:fofType)=> (Xphi Xx)))->((exu (fun (Xx:fofType)=> (Xpsi Xx)))->(((eq fofType) (descr (fun (Xx:fofType)=> (Xphi Xx)))) (descr (fun (Xx:fofType)=> (Xpsi Xx)))))))):Prop.
% Definition dsetconstr__Cong:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->(forall (Xphi:(fofType->Prop)) (Xpsi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((((eq fofType) Xx) Xy)->((iff (Xphi Xx)) (Xpsi Xy)))))))->(((eq fofType) ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) ((dsetconstr B) (fun (Xx:fofType)=> (Xpsi Xx)))))))):Prop.
% Parameter subset:(fofType->(fofType->Prop)).
% Parameter disjoint:(fofType->(fofType->Prop)).
% Parameter setsmeet:(fofType->(fofType->Prop)).
% Definition subsetI1:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((subset A) B))):Prop.
% Definition eqimpsubset2:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->((subset B) A))):Prop.
% Definition eqimpsubset1:=(forall (A:fofType) (B:fofType), ((((eq fofType) A) B)->((subset A) B))):Prop.
% Definition subsetI2:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((subset A) B))):Prop.
% Definition emptysetsubset:=(forall (A:fofType), ((subset emptyset) A)):Prop.
% Definition subsetE:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((subset A) B)->(((in Xx) A)->((in Xx) B)))):Prop.
% Definition subsetE2:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((subset A) B)->((((in Xx) B)->False)->(((in Xx) A)->False)))):Prop.
% Definition notsubsetI:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->(((subset A) B)->False)))):Prop.
% Definition notequalI1:=(forall (A:fofType) (B:fofType), ((((subset A) B)->False)->(not (((eq fofType) A) B)))):Prop.
% Definition notequalI2:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->(not (((eq fofType) A) B))))):Prop.
% Definition subsetRefl:=(forall (A:fofType), ((subset A) A)):Prop.
% Definition subsetTrans:=(forall (A:fofType) (B:fofType) (C:fofType), (((subset A) B)->(((subset B) C)->((subset A) C)))):Prop.
% Definition setadjoinSub:=(forall (Xx:fofType) (A:fofType), ((subset A) ((setadjoin Xx) A))):Prop.
% Definition setadjoinSub2:=(forall (A:fofType) (Xx:fofType) (B:fofType), (((subset A) B)->((subset A) ((setadjoin Xx) B)))):Prop.
% Definition subset2powerset:=(forall (A:fofType) (B:fofType), (((subset A) B)->((in A) (powerset B)))):Prop.
% Definition setextsub:=(forall (A:fofType) (B:fofType), (((subset A) B)->(((subset B) A)->(((eq fofType) A) B)))):Prop.
% Definition subsetemptysetimpeq:=(forall (A:fofType), (((subset A) emptyset)->(((eq fofType) A) emptyset))):Prop.
% Definition powersetI1:=(forall (A:fofType) (B:fofType), (((subset B) A)->((in B) (powerset A)))):Prop.
% Definition powersetE1:=(forall (A:fofType) (B:fofType), (((in B) (powerset A))->((subset B) A))):Prop.
% Definition inPowerset:=(forall (A:fofType), ((in A) (powerset A))):Prop.
% Definition powersetsubset:=(forall (A:fofType) (B:fofType), (((subset A) B)->((subset (powerset A)) (powerset B)))):Prop.
% Definition sepInPowerset:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((in ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) (powerset A))):Prop.
% Definition sepSubset:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((subset ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx)))) A)):Prop.
% Parameter binunion:(fofType->(fofType->fofType)).
% Definition binunionIL:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((in Xx) ((binunion A) B)))):Prop.
% Definition upairset2IR:=(forall (Xx:fofType) (Xy:fofType), ((in Xy) ((setadjoin Xx) ((setadjoin Xy) emptyset)))):Prop.
% Definition binunionIR:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) B)->((in Xx) ((binunion A) B)))):Prop.
% Definition binunionEcases:=(forall (A:fofType) (B:fofType) (Xx:fofType) (Xphi:Prop), (((in Xx) ((binunion A) B))->((((in Xx) A)->Xphi)->((((in Xx) B)->Xphi)->Xphi)))):Prop.
% Definition binunionE:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((binunion A) B))->((or ((in Xx) A)) ((in Xx) B)))):Prop.
% Definition binunionLsub:=(forall (A:fofType) (B:fofType), ((subset A) ((binunion A) B))):Prop.
% Definition binunionRsub:=(forall (A:fofType) (B:fofType), ((subset B) ((binunion A) B))):Prop.
% Parameter binintersect:(fofType->(fofType->fofType)).
% Definition binintersectI:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(((in Xx) B)->((in Xx) ((binintersect A) B))))):Prop.
% Definition binintersectSubset5:=(forall (A:fofType) (B:fofType) (C:fofType), (((subset C) A)->(((subset C) B)->((subset C) ((binintersect A) B))))):Prop.
% Definition binintersectEL:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((binintersect A) B))->((in Xx) A))):Prop.
% Definition binintersectLsub:=(forall (A:fofType) (B:fofType), ((subset ((binintersect A) B)) A)):Prop.
% Definition binintersectSubset2:=(forall (A:fofType) (B:fofType), (((subset A) B)->(((eq fofType) ((binintersect A) B)) A))):Prop.
% Definition binintersectSubset3:=(forall (A:fofType) (B:fofType), ((((eq fofType) ((binintersect A) B)) B)->((subset B) A))):Prop.
% Definition binintersectER:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((binintersect A) B))->((in Xx) B))):Prop.
% Definition disjointsetsI1:=(forall (A:fofType) (B:fofType), ((((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((in Xx) B))))->False)->(((eq fofType) ((binintersect A) B)) emptyset))):Prop.
% Definition binintersectRsub:=(forall (A:fofType) (B:fofType), ((subset ((binintersect A) B)) B)):Prop.
% Definition binintersectSubset4:=(forall (A:fofType) (B:fofType), (((subset B) A)->(((eq fofType) ((binintersect A) B)) B))):Prop.
% Definition binintersectSubset1:=(forall (A:fofType) (B:fofType), ((((eq fofType) ((binintersect A) B)) A)->((subset A) B))):Prop.
% Definition bs114d:=(forall (A:fofType) (B:fofType) (C:fofType), (((eq fofType) ((binintersect A) ((binunion B) C))) ((binunion ((binintersect A) B)) ((binintersect A) C)))):Prop.
% Parameter regular:(fofType->Prop).
% Parameter setminus:(fofType->(fofType->fofType)).
% Definition setminusI:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->((in Xx) ((setminus A) B))))):Prop.
% Definition setminusEL:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((setminus A) B))->((in Xx) A))):Prop.
% Definition setminusER:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((setminus A) B))->(((in Xx) B)->False))):Prop.
% Definition setminusSubset2:=(forall (A:fofType) (B:fofType), (((subset A) B)->(((eq fofType) ((setminus A) B)) emptyset))):Prop.
% Definition setminusERneg:=(forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) ((setminus A) B))->False)->(((in Xx) A)->((in Xx) B)))):Prop.
% Definition setminusELneg:=(forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) ((setminus A) B))->False)->((((in Xx) B)->False)->(((in Xx) A)->False)))):Prop.
% Definition setminusILneg:=(forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) A)->False)->(((in Xx) ((setminus A) B))->False))):Prop.
% Definition setminusIRneg:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) B)->(((in Xx) ((setminus A) B))->False))):Prop.
% Definition setminusLsub:=(forall (A:fofType) (B:fofType), ((subset ((setminus A) B)) A)):Prop.
% Definition setminusSubset1:=(forall (A:fofType) (B:fofType), ((((eq fofType) ((setminus A) B)) emptyset)->((subset A) B))):Prop.
% Parameter symdiff:(fofType->(fofType->fofType)).
% Definition symdiffE:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) ((symdiff A) B))->(forall (Xphi:Prop), ((((in Xx) A)->((((in Xx) B)->False)->Xphi))->(((((in Xx) A)->False)->(((in Xx) B)->Xphi))->Xphi))))):Prop.
% Definition symdiffI1:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((((in Xx) B)->False)->((in Xx) ((symdiff A) B))))):Prop.
% Definition symdiffI2:=(forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) A)->False)->(((in Xx) B)->((in Xx) ((symdiff A) B))))):Prop.
% Definition symdiffIneg1:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(((in Xx) B)->(((in Xx) ((symdiff A) B))->False)))):Prop.
% Definition symdiffIneg2:=(forall (A:fofType) (B:fofType) (Xx:fofType), ((((in Xx) A)->False)->((((in Xx) B)->False)->(((in Xx) ((symdiff A) B))->False)))):Prop.
% Parameter iskpair:(fofType->Prop).
% Definition secondinupair:=(forall (Xx:fofType) (Xy:fofType), ((in Xy) ((setadjoin Xx) ((setadjoin Xy) emptyset)))):Prop.
% Definition setukpairIL:=(forall (Xx:fofType) (Xy:fofType), ((in Xx) (setunion ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))))):Prop.
% Definition setukpairIR:=(forall (Xx:fofType) (Xy:fofType), ((in Xy) (setunion ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))))):Prop.
% Definition kpairiskpair:=(forall (Xx:fofType) (Xy:fofType), (iskpair ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))):Prop.
% Parameter kpair:(fofType->(fofType->fofType)).
% Definition kpairp:=(forall (Xx:fofType) (Xy:fofType), (iskpair ((kpair Xx) Xy))):Prop.
% Parameter cartprod:(fofType->(fofType->fofType)).
% Definition singletonsubset:=(forall (A:fofType) (Xx:fofType), (((in Xx) A)->((subset ((setadjoin Xx) emptyset)) A))):Prop.
% Definition singletoninpowerset:=(forall (A:fofType) (Xx:fofType), (((in Xx) A)->((in ((setadjoin Xx) emptyset)) (powerset A)))):Prop.
% Definition singletoninpowunion:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->((in ((setadjoin Xx) emptyset)) (powerset ((binunion A) B))))):Prop.
% Definition upairset2E:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((in Xz) ((setadjoin Xx) ((setadjoin Xy) emptyset)))->((or (((eq fofType) Xz) Xx)) (((eq fofType) Xz) Xy)))):Prop.
% Definition upairsubunion:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((subset ((setadjoin Xx) ((setadjoin Xy) emptyset))) ((binunion A) B)))))):Prop.
% Definition upairinpowunion:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((setadjoin Xx) ((setadjoin Xy) emptyset))) (powerset ((binunion A) B))))))):Prop.
% Definition ubforcartprodlem1:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((subset ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) (powerset ((binunion A) B))))))):Prop.
% Definition ubforcartprodlem2:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) (powerset (powerset ((binunion A) B)))))))):Prop.
% Definition ubforcartprodlem3:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))))))):Prop.
% Definition cartprodpairin:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))):Prop.
% Definition cartprodmempair1:=(forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) B)) (((eq fofType) Xu) ((kpair Xx) Xy)))))))))):Prop.
% Definition cartprodmempair:=(forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->(iskpair Xu))):Prop.
% Definition setunionE2:=(forall (A:fofType) (Xx:fofType), (((in Xx) (setunion A))->((ex fofType) (fun (X:fofType)=> ((and ((in X) A)) ((in Xx) X)))))):Prop.
% Definition setunionsingleton1:=(forall (A:fofType), ((subset (setunion ((setadjoin A) emptyset))) A)):Prop.
% Definition setunionsingleton2:=(forall (A:fofType), ((subset A) (setunion ((setadjoin A) emptyset)))):Prop.
% Definition setunionsingleton:=(forall (Xx:fofType), (((eq fofType) (setunion ((setadjoin Xx) emptyset))) Xx)):Prop.
% Parameter singleton:(fofType->Prop).
% Definition singletonprop:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))))->(((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->(singleton ((dsetconstr A) (fun (Xx:fofType)=> (Xphi Xx))))))):Prop.
% Parameter ex1:(fofType->((fofType->Prop)->Prop)).
% Definition ex1E1:=(forall (A:fofType) (Xphi:(fofType->Prop)), (((ex1 A) (fun (Xx:fofType)=> (Xphi Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx)))))):Prop.
% Definition ex1I:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((forall (Xy:fofType), (((in Xy) A)->((Xphi Xy)->(((eq fofType) Xy) Xx))))->((ex1 A) (fun (Xy:fofType)=> (Xphi Xy))))))):Prop.
% Definition ex1I2:=(forall (A:fofType) (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy)))))))->(((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) A)) (Xphi Xx))))->((ex1 A) (fun (Xx:fofType)=> (Xphi Xx)))))):Prop.
% Definition singletonsuniq:=(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) ((setadjoin Xx) emptyset)) ((setadjoin Xy) emptyset))->(((eq fofType) Xx) Xy))):Prop.
% Parameter atmost1p:(fofType->Prop).
% Parameter atleast2p:(fofType->Prop).
% Parameter atmost2p:(fofType->Prop).
% Parameter upairsetp:(fofType->Prop).
% Definition setukpairinjL1:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((in ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))->(((eq fofType) Xx) Xz))):Prop.
% Definition kfstsingleton:=(forall (Xu:fofType), ((iskpair Xu)->(singleton ((dsetconstr (setunion Xu)) (fun (Xx:fofType)=> ((in ((setadjoin Xx) emptyset)) Xu)))))):Prop.
% Definition theprop:=(forall (X:fofType), ((singleton X)->((in (setunion X)) X))):Prop.
% Parameter kfst:(fofType->fofType).
% Definition kfstpairEq:=(forall (Xx:fofType) (Xy:fofType), (((eq fofType) (kfst ((kpair Xx) Xy))) Xx)):Prop.
% Definition cartprodfstin:=(forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->((in (kfst Xu)) A))):Prop.
% Definition setukpairinjL2:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xz) ((setadjoin Xu) emptyset))) emptyset)))->(((eq fofType) Xx) Xz))):Prop.
% Definition setukpairinjL:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((kpair Xx) Xy)) ((kpair Xz) Xu))->(((eq fofType) Xx) Xz))):Prop.
% Definition setukpairinjR11:=(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(((eq fofType) ((setadjoin Xx) ((setadjoin Xy) emptyset))) ((setadjoin Xx) emptyset)))):Prop.
% Definition setukpairinjR12:=(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) Xx) Xy)->(((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xx) emptyset)) emptyset)))):Prop.
% Definition setukpairinjR1:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xz) ((setadjoin Xu) emptyset))) emptyset)))->((((eq fofType) Xz) Xu)->(((eq fofType) Xy) Xu)))):Prop.
% Definition upairequniteq:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), ((((eq fofType) ((setadjoin Xx) ((setadjoin Xy) emptyset))) ((setadjoin Xz) emptyset))->(((eq fofType) Xx) Xy))):Prop.
% Definition setukpairinjR2:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) ((setadjoin ((setadjoin Xz) emptyset)) ((setadjoin ((setadjoin Xz) ((setadjoin Xu) emptyset))) emptyset)))->(((eq fofType) Xy) Xu))):Prop.
% Definition setukpairinjR:=(forall (Xx:fofType) (Xy:fofType) (Xz:fofType) (Xu:fofType), ((((eq fofType) ((kpair Xx) Xy)) ((kpair Xz) Xu))->(((eq fofType) Xy) Xu))):Prop.
% Definition ksndsingleton:=(forall (Xu:fofType), ((iskpair Xu)->(singleton ((dsetconstr (setunion Xu)) (fun (Xx:fofType)=> (((eq fofType) Xu) ((kpair (kfst Xu)) Xx))))))):Prop.
% Parameter ksnd:(fofType->fofType).
% Definition ksndpairEq:=(forall (Xx:fofType) (Xy:fofType), (((eq fofType) (ksnd ((kpair Xx) Xy))) Xy)):Prop.
% Definition kpairsurjEq:=(forall (Xu:fofType), ((iskpair Xu)->(((eq fofType) ((kpair (kfst Xu)) (ksnd Xu))) Xu))):Prop.
% Definition cartprodsndin:=(forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->((in (ksnd Xu)) B))):Prop.
% Definition cartprodpairmemEL:=(forall (A:fofType) (B:fofType) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) ((cartprod A) B))->((in Xx) A))):Prop.
% Definition cartprodpairmemER:=(forall (A:fofType) (B:fofType) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) ((cartprod A) B))->((in Xy) B))):Prop.
% Definition cartprodmempaircEq:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy)))))):Prop.
% Definition cartprodfstpairEq:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((eq fofType) (kfst ((kpair Xx) Xy))) Xx))))):Prop.
% Definition cartprodsndpairEq:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((eq fofType) (ksnd ((kpair Xx) Xy))) Xy))))):Prop.
% Definition cartprodpairsurjEq:=(forall (A:fofType) (B:fofType) (Xu:fofType), (((in Xu) ((cartprod A) B))->(((eq fofType) ((kpair (kfst Xu)) (ksnd Xu))) Xu))):Prop.
% Parameter breln:(fofType->(fofType->(fofType->Prop))).
% Parameter dpsetconstr:(fofType->(fofType->((fofType->(fofType->Prop))->fofType))).
% Definition dpsetconstrI:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((Xphi Xx) Xy)->((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))))))):Prop.
% Definition dpsetconstrSub:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))), ((subset (((dpsetconstr A) B) (fun (Xx:fofType) (Xy:fofType)=> ((Xphi Xx) Xy)))) ((cartprod A) B))):Prop.
% Definition setOfPairsIsBReln:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))), (((breln A) B) (((dpsetconstr A) B) (fun (Xx:fofType) (Xy:fofType)=> ((Xphi Xx) Xy))))):Prop.
% Definition dpsetconstrERa:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((Xphi Xx) Xy)))))):Prop.
% Definition dpsetconstrEL1:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((in Xx) A))):Prop.
% Definition dpsetconstrEL2:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((in Xy) B))):Prop.
% Definition dpsetconstrER:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))) (Xx:fofType) (Xy:fofType), (((in ((kpair Xx) Xy)) (((dpsetconstr A) B) (fun (Xz:fofType) (Xu:fofType)=> ((Xphi Xz) Xu))))->((Xphi Xx) Xy))):Prop.
% Parameter func:(fofType->(fofType->(fofType->Prop))).
% Parameter funcSet:(fofType->(fofType->fofType)).
% Definition funcImageSingleton:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->(singleton ((dsetconstr B) (fun (Xy:fofType)=> ((in ((kpair Xx) Xy)) Xf)))))))):Prop.
% Definition apProp:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->((in (setunion ((dsetconstr B) (fun (Xy:fofType)=> ((in ((kpair Xx) Xy)) Xf))))) B))))):Prop.
% Parameter ap:(fofType->(fofType->(fofType->(fofType->fofType)))).
% Definition app:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->((in ((((ap A) B) Xf) Xx)) B))))):Prop.
% Definition infuncsetfunc:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(((func A) B) Xf))):Prop.
% Definition ap2p:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->((in ((((ap A) B) Xf) Xx)) B))))):Prop.
% Definition funcinfuncset:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->((in Xf) ((funcSet A) B)))):Prop.
% Definition lamProp:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(((func A) B) (((dpsetconstr A) B) (fun (Xx:fofType) (Xy:fofType)=> (((eq fofType) (Xf Xx)) Xy)))))):Prop.
% Parameter lam:(fofType->(fofType->((fofType->fofType)->fofType))).
% Definition lamp:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(((func A) B) (((lam A) B) (fun (Xx:fofType)=> (Xf Xx)))))):Prop.
% Definition lam2p:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->((in (((lam A) B) (fun (Xx:fofType)=> (Xf Xx)))) ((funcSet A) B)))):Prop.
% Definition brelnall1:=(forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx))))))):Prop.
% Definition brelnall2:=(forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx))))))):Prop.
% Definition ex1E2:=(forall (A:fofType) (Xphi:(fofType->Prop)), (((ex1 A) (fun (Xx:fofType)=> (Xphi Xx)))->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi Xx)->((Xphi Xy)->(((eq fofType) Xx) Xy))))))))):Prop.
% Definition funcGraphProp1:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->((in ((kpair Xx) ((((ap A) B) Xf) Xx))) Xf))))):Prop.
% Definition funcGraphProp3:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->((in ((kpair Xx) ((((ap A) B) Xf) Xx))) Xf))))):Prop.
% Definition funcGraphProp2:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) Xf)->(((eq fofType) ((((ap A) B) Xf) Xx)) Xy)))))))):Prop.
% Definition funcextLem:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xg:fofType), ((((func A) B) Xg)->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xg) Xx))))->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) Xg)->((in ((kpair Xx) Xy)) Xf))))))))))):Prop.
% Definition funcGraphProp4:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) Xf)->(((eq fofType) ((((ap A) B) Xf) Xx)) Xy)))))))):Prop.
% Definition subbreln:=(forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (S:fofType), ((((breln A) B) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((subset R) S)))))):Prop.
% Definition eqbreln:=(forall (A:fofType) (B:fofType) (R:fofType), ((((breln A) B) R)->(forall (S:fofType), ((((breln A) B) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) R))))))->(((eq fofType) R) S))))))):Prop.
% Definition funcext:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xg:fofType), ((((func A) B) Xg)->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xg) Xx))))->(((eq fofType) Xf) Xg)))))):Prop.
% Definition funcext2:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xg:fofType), (((in Xg) ((funcSet A) B))->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xg) Xx))))->(((eq fofType) Xf) Xg)))))):Prop.
% Definition ap2apEq1:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xf) Xx)))))):Prop.
% Definition ap2apEq2:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) Xf) Xx)) ((((ap A) B) Xf) Xx)))))):Prop.
% Definition beta1:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) (((lam A) B) (fun (Xy:fofType)=> (Xf Xy)))) Xx)) (Xf Xx)))))):Prop.
% Definition eta1:=(forall (A:fofType) (B:fofType) (Xf:fofType), ((((func A) B) Xf)->(((eq fofType) (((lam A) B) (fun (Xx:fofType)=> ((((ap A) B) Xf) Xx)))) Xf))):Prop.
% Definition lam2lamEq:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(((eq fofType) (((lam A) B) (fun (Xx:fofType)=> (Xf Xx)))) (((lam A) B) (fun (Xx:fofType)=> (Xf Xx)))))):Prop.
% Definition beta2:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap A) B) (((lam A) B) (fun (Xy:fofType)=> (Xf Xy)))) Xx)) (Xf Xx)))))):Prop.
% Definition eta2:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->(((eq fofType) (((lam A) B) (fun (Xx:fofType)=> ((((ap A) B) Xf) Xx)))) Xf))):Prop.
% Definition iffalseProp1:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi->False)->((in Xy) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy))))))))))):Prop.
% Definition iffalseProp2:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi->False)->(((eq fofType) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))) ((setadjoin Xy) emptyset))))))):Prop.
% Definition iftrueProp1:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(Xphi->((in Xx) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy))))))))))):Prop.
% Definition iftrueProp2:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(Xphi->(((eq fofType) ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))) ((setadjoin Xx) emptyset))))))):Prop.
% Definition ifSingleton:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(singleton ((dsetconstr A) (fun (Xz:fofType)=> ((or ((and Xphi) (((eq fofType) Xz) Xx))) ((and (Xphi->False)) (((eq fofType) Xz) Xy)))))))))):Prop.
% Parameter if:(fofType->(Prop->(fofType->(fofType->fofType)))).
% Definition ifp:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((in ((((if A) Xphi) Xx) Xy)) A))))):Prop.
% Definition theeq:=(forall (X:fofType), ((singleton X)->(forall (Xx:fofType), (((in Xx) X)->(((eq fofType) (setunion X)) Xx))))):Prop.
% Definition iftrue:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(Xphi->(((eq fofType) ((((if A) Xphi) Xx) Xy)) Xx)))))):Prop.
% Definition iffalse:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((Xphi->False)->(((eq fofType) ((((if A) Xphi) Xx) Xy)) Xy)))))):Prop.
% Definition iftrueorfalse:=(forall (A:fofType) (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->((in ((((if A) Xphi) Xx) Xy)) ((setadjoin Xx) ((setadjoin Xy) emptyset))))))):Prop.
% Definition binintersectT_lem:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((binintersect X) Y)) (powerset A)))))):Prop.
% Definition binunionT_lem:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((binunion X) Y)) (powerset A)))))):Prop.
% Definition powersetT_lem:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->((in (powerset X)) (powerset (powerset A))))):Prop.
% Definition setminusT_lem:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus X) Y)) (powerset A)))))):Prop.
% Definition complementT_lem:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->((in ((setminus A) X)) (powerset A)))):Prop.
% Definition setextT:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) Y))))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) Y)->((in Xx) X))))->(((eq fofType) X) Y))))))):Prop.
% Definition subsetTI:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) Y))))->((subset X) Y)))))):Prop.
% Definition powersetTI1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) Y))))->((in X) (powerset Y))))))):Prop.
% Definition powersetTE1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in X) (powerset Y))->(((in Xx) X)->((in Xx) Y))))))))):Prop.
% Definition complementTI1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->(((in Xx) ((setminus A) X))->False)))))):Prop.
% Definition complementTE1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) ((setminus A) X))->False)->((in Xx) X)))))):Prop.
% Definition binintersectTELcontra:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) X)->False)->(((in Xx) ((binintersect X) Y))->False)))))))):Prop.
% Definition binintersectTERcontra:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) Y)->False)->(((in Xx) ((binintersect X) Y))->False)))))))):Prop.
% Definition contrasubsetT:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((subset X) ((setminus A) Y))->(((in Xx) Y)->(((in Xx) X)->False))))))))):Prop.
% Definition contrasubsetT1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((subset X) Y)->((((in Xx) Y)->False)->(((in Xx) X)->False))))))))):Prop.
% Definition contrasubsetT2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) Y)->((subset ((setminus A) Y)) ((setminus A) X))))))):Prop.
% Definition contrasubsetT3:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset ((setminus A) Y)) ((setminus A) X))->((subset X) Y)))))):Prop.
% Definition doubleComplementI1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) X)->((in Xx) ((setminus A) ((setminus A) X)))))))):Prop.
% Definition doubleComplementE1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((setminus A) X)))->((in Xx) X)))))):Prop.
% Definition doubleComplementSub1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->((subset X) ((setminus A) ((setminus A) X))))):Prop.
% Definition doubleComplementSub2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->((subset ((setminus A) ((setminus A) X))) X))):Prop.
% Definition doubleComplementEq:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(((eq fofType) X) ((setminus A) ((setminus A) X))))):Prop.
% Definition complementTnotintersectT:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) X))->(((in Xx) ((binintersect X) Y))->False)))))))):Prop.
% Definition complementImpComplementIntersect:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) X))->((in Xx) ((setminus A) ((binintersect X) Y)))))))))):Prop.
% Definition complementSubsetComplementIntersect:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((subset ((setminus A) X)) ((setminus A) ((binintersect X) Y))))))):Prop.
% Definition complementInPowersetComplementIntersect:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) X)) (powerset ((setminus A) ((binintersect X) Y)))))))):Prop.
% Definition contraSubsetComplement:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) ((setminus A) Y))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) Y)->((in Xx) ((setminus A) X)))))))))):Prop.
% Definition complementTcontraSubset:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) ((setminus A) Y))->((subset Y) ((setminus A) X))))))):Prop.
% Definition binunionTILcontra:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) ((binunion X) Y))->False)->(((in Xx) X)->False)))))))):Prop.
% Definition binunionTIRcontra:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) ((binunion X) Y))->False)->(((in Xx) Y)->False)))))))):Prop.
% Definition inIntersectImpInUnion:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect X) Y))->((in Xx) ((binunion X) Z))))))))))):Prop.
% Definition inIntersectImpInUnion2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect X) Y))->((in Xx) ((binunion Y) Z))))))))))):Prop.
% Definition inIntersectImpInIntersectUnions:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect X) Y))->((in Xx) ((binintersect ((binunion X) Z)) ((binunion Y) Z)))))))))))):Prop.
% Definition intersectInPowersetIntersectUnions:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->((in ((binintersect X) Y)) (powerset ((binintersect ((binunion X) Z)) ((binunion Y) Z)))))))))):Prop.
% Definition inComplementUnionImpNotIn1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->(((in Xx) X)->False)))))))):Prop.
% Definition inComplementUnionImpInComplement1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((setminus A) X))))))))):Prop.
% Definition binunionTE:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xphi:Prop) (Xx:fofType), (((in Xx) A)->(((in Xx) ((binunion X) Y))->((((in Xx) X)->Xphi)->((((in Xx) Y)->Xphi)->Xphi))))))))):Prop.
% Definition binunionTEcontra:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->((((in Xx) X)->False)->((((in Xx) Y)->False)->(((in Xx) ((binunion X) Y))->False))))))))):Prop.
% Definition demorgan2a1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((setminus A) X))))))))):Prop.
% Definition complementUnionInPowersetComplement:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) ((binunion X) Y))) (powerset ((setminus A) X))))))):Prop.
% Definition demorgan2a2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((setminus A) Y))))))))):Prop.
% Definition demorgan1a:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binintersect X) Y)))->((in Xx) ((binunion ((setminus A) X)) ((setminus A) Y)))))))))):Prop.
% Definition demorgan1b:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binunion ((setminus A) X)) ((setminus A) Y)))->((in Xx) ((setminus A) ((binintersect X) Y)))))))))):Prop.
% Definition demorgan1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((eq fofType) ((setminus A) ((binintersect X) Y))) ((binunion ((setminus A) X)) ((setminus A) Y))))))):Prop.
% Definition demorgan2a:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) ((binunion X) Y)))->((in Xx) ((binintersect ((setminus A) X)) ((setminus A) Y)))))))))):Prop.
% Definition demorgan2b2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((setminus A) X))->(((in Xx) ((setminus A) Y))->((in Xx) ((setminus A) ((binunion X) Y))))))))))):Prop.
% Definition demorgan2b:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) A)->(((in Xx) ((binintersect ((setminus A) X)) ((setminus A) Y)))->((in Xx) ((setminus A) ((binunion X) Y)))))))))):Prop.
% Definition demorgan2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((eq fofType) ((setminus A) ((binunion X) Y))) ((binintersect ((setminus A) X)) ((setminus A) Y))))))):Prop.
% Definition woz13rule0:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Xx:fofType), (((in Xx) ((binintersect X) Y))->((in Xx) A))))))):Prop.
% Definition woz13rule1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(((subset X) Z)->((subset ((binintersect X) Y)) Z)))))))):Prop.
% Definition woz13rule2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(((subset Y) Z)->((subset ((binintersect X) Y)) Z)))))))):Prop.
% Definition woz13rule3:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(((subset X) Y)->(((subset X) Z)->((subset X) ((binintersect Y) Z)))))))))):Prop.
% Definition woz13rule4:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (W:fofType), (((in W) (powerset A))->(((subset X) Z)->(((subset Y) W)->((subset ((binintersect X) Y)) ((binintersect Z) W)))))))))))):Prop.
% Definition woz1_1:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) X)) (powerset ((setminus A) ((binintersect X) Y)))))))):Prop.
% Definition woz1_2:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->(forall (W:fofType), (((in W) (powerset A))->(((eq fofType) ((setminus A) ((binintersect ((binunion X) Y)) ((binunion Z) W)))) ((binunion ((binintersect ((setminus A) X)) ((setminus A) Y))) ((binintersect ((setminus A) Z)) ((setminus A) W)))))))))))):Prop.
% Definition woz1_3:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(forall (Z:fofType), (((in Z) (powerset A))->((in ((binintersect X) Y)) (powerset ((binintersect ((binunion X) Z)) ((binunion Y) Z)))))))))):Prop.
% Definition woz1_4:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->(((subset X) ((setminus A) Y))->((subset Y) ((setminus A) X))))))):Prop.
% Definition woz1_5:=(forall (A:fofType) (X:fofType), (((in X) (powerset A))->(forall (Y:fofType), (((in Y) (powerset A))->((in ((setminus A) ((binunion X) Y))) (powerset ((setminus A) X))))))):Prop.
% Parameter breln1:(fofType->(fofType->Prop)).
% Definition breln1all2:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx))))))):Prop.
% Parameter breln1Set:(fofType->fofType).
% Definition breln1SetBreln1:=(forall (A:fofType) (R:fofType), (((in R) (breln1Set A))->((breln1 A) R))):Prop.
% Parameter transitive:(fofType->(fofType->Prop)).
% Parameter antisymmetric:(fofType->(fofType->Prop)).
% Parameter reflexive:(fofType->(fofType->Prop)).
% Parameter refltransitive:(fofType->(fofType->Prop)).
% Parameter refllinearorder:(fofType->(fofType->Prop)).
% Parameter reflwellordering:(fofType->(fofType->Prop)).
% Definition choice2fnsingleton:=(forall (A:fofType) (B:fofType) (Xphi:(fofType->(fofType->Prop))), ((forall (Xx:fofType), (((in Xx) A)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) B)) ((Xphi Xx) Xy))))))->(forall (R:fofType), (((in R) (breln1Set B))->(((reflwellordering B) R)->(forall (Xx:fofType), (((in Xx) A)->(singleton ((dsetconstr B) (fun (Xy:fofType)=> ((and ((Xphi Xx) Xy)) (forall (Xz:fofType), (((in Xz) B)->(((Xphi Xx) Xz)->((in ((kpair Xy) Xz)) R))))))))))))))):Prop.
% Definition setOfPairsIsBReln1:=(forall (A:fofType) (Xphi:(fofType->(fofType->Prop))), ((breln1 A) (((dpsetconstr A) A) (fun (Xx:fofType) (Xy:fofType)=> ((Xphi Xx) Xy))))):Prop.
% Definition breln1all1:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xphi:(fofType->Prop)), ((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->(Xphi ((kpair Xx) Xy)))))))->(forall (Xx:fofType), (((in Xx) R)->(Xphi Xx))))))):Prop.
% Definition subbreln1:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((subset R) S)))))):Prop.
% Definition eqbreln1:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) S))))))->((forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) R))))))->(((eq fofType) R) S))))))):Prop.
% Parameter breln1invset:(fofType->(fofType->fofType)).
% Definition breln1invprop:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->((breln1 A) ((breln1invset A) R)))):Prop.
% Definition breln1invI:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xy) Xx)) ((breln1invset A) R))))))))):Prop.
% Definition breln1invE:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xy) Xx)) ((breln1invset A) R))->((in ((kpair Xx) Xy)) R)))))))):Prop.
% Parameter breln1compset:(fofType->(fofType->(fofType->fofType))).
% Definition breln1compprop:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((breln1 A) (((breln1compset A) R) S)))))):Prop.
% Definition breln1compI:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(forall (Xz:fofType), (((in Xz) A)->(((in ((kpair Xx) Xz)) R)->(((in ((kpair Xz) Xy)) S)->((in ((kpair Xx) Xy)) (((breln1compset A) R) S)))))))))))))):Prop.
% Definition breln1compE:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) (((breln1compset A) R) S))->((ex fofType) (fun (Xz:fofType)=> ((and ((and ((in Xz) A)) ((in ((kpair Xx) Xz)) R))) ((in ((kpair Xz) Xy)) S))))))))))))):Prop.
% Definition breln1compEex:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) (((breln1compset A) R) S))->(forall (Xphi:Prop), ((forall (Xz:fofType), (((in Xz) A)->(((in ((kpair Xx) Xz)) R)->(((in ((kpair Xz) Xy)) S)->Xphi))))->Xphi))))))))))):Prop.
% Definition breln1unionprop:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->((breln1 A) ((binunion R) S)))))):Prop.
% Definition breln1unionIL:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))):Prop.
% Definition breln1unionIR:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))):Prop.
% Definition breln1unionI:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))):Prop.
% Definition breln1unionE:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) ((binunion R) S))->((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))))))))))):Prop.
% Definition breln1unionEcases:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) ((binunion R) S))->(forall (Xphi:Prop), ((((in ((kpair Xx) Xy)) R)->Xphi)->((((in ((kpair Xx) Xy)) S)->Xphi)->Xphi)))))))))))):Prop.
% Definition breln1unionCommutes:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(((eq fofType) ((binunion R) S)) ((binunion S) R)))))):Prop.
% Definition woz2Ex:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(((eq fofType) R) ((breln1invset A) ((breln1invset A) R))))):Prop.
% Definition woz2W:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(((eq fofType) ((breln1invset A) (((breln1compset A) R) S))) (((breln1compset A) ((breln1invset A) S)) ((breln1invset A) R))))))):Prop.
% Definition woz2A:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (T:fofType), (((breln1 A) T)->(((eq fofType) (((breln1compset A) ((binunion R) S)) T)) ((binunion (((breln1compset A) R) T)) (((breln1compset A) S) T))))))))):Prop.
% Definition woz2B:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (T:fofType), (((breln1 A) T)->(((eq fofType) (((breln1compset A) ((binunion R) S)) T)) ((binunion ((breln1invset A) (((breln1compset A) ((breln1invset A) T)) ((breln1invset A) S)))) ((breln1invset A) (((breln1compset A) ((breln1invset A) T)) ((breln1invset A) R))))))))))):Prop.
% Definition image1Ex:=(forall (A:fofType) (Xf:(fofType->fofType)), ((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))))):Prop.
% Definition image1Ex1:=(forall (A:fofType) (Xf:(fofType->fofType)), (exu (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))))):Prop.
% Parameter image1:(fofType->((fofType->fofType)->fofType)).
% Definition image1Equiv:=(forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), ((iff ((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))):Prop.
% Definition image1E:=(forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), (((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))):Prop.
% Definition image1I:=(forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))->((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))):Prop.
% Parameter injective:(fofType->(fofType->(fofType->Prop))).
% Parameter injFuncSet:(fofType->(fofType->fofType)).
% Definition injFuncInInjFuncSet:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((funcSet A) B))->((((injective A) B) Xf)->((in Xf) ((injFuncSet A) B))))):Prop.
% Definition injFuncSetFuncIn:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((injFuncSet A) B))->((in Xf) ((funcSet A) B)))):Prop.
% Definition injFuncSetFuncInj:=(forall (Xx:fofType) (Xy:fofType) (Xf:fofType), (((in Xf) ((injFuncSet Xx) Xy))->(((injective Xx) Xy) Xf))):Prop.
% Parameter surjective:(fofType->(fofType->(fofType->Prop))).
% Parameter surjFuncSet:(fofType->(fofType->fofType)).
% Definition surjFuncSetFuncIn:=(forall (A:fofType) (B:fofType) (Xf:fofType), (((in Xf) ((surjFuncSet A) B))->((in Xf) ((funcSet A) B)))):Prop.
% Definition surjFuncSetFuncSurj:=(forall (Xx:fofType) (Xy:fofType) (Xf:fofType), (((in Xf) ((surjFuncSet Xx) Xy))->(((surjective Xx) Xy) Xf))):Prop.
% Definition leftInvIsSurj:=(forall (A:fofType) (B:fofType) (Xf:(fofType->fofType)), ((forall (Xx:fofType), (((in Xx) A)->((in (Xf Xx)) B)))->(forall (Xg:fofType), (((in Xg) ((funcSet B) A))->((forall (Xx:fofType), (((in Xx) A)->(((eq fofType) ((((ap B) A) Xg) (Xf Xx))) Xx)))->(((surjective B) A) Xg)))))):Prop.
% Definition surjCantorThm:=(forall (A:fofType) (Xf:fofType), (((in Xf) ((funcSet A) (powerset A)))->((((surjective A) (powerset A)) Xf)->False))):Prop.
% Definition foundation2:=(forall (A:fofType), ((nonempty A)->((ex fofType) (fun (X:fofType)=> ((and ((in X) A)) (((eq fofType) ((binintersect X) A)) emptyset)))))):Prop.
% Definition notinself:=(forall (A:fofType), (((in A) A)->False)):Prop.
% Definition notinself2:=(forall (A:fofType) (B:fofType), (((in A) B)->(((in B) A)->False))):Prop.
% Parameter omegaS:(fofType->fofType).
% Definition omegaSp:=(forall (Xx:fofType), (((in Xx) omega)->((in (omegaS Xx)) omega))):Prop.
% Definition omegaSclos:=(forall (Xx:fofType), (((in Xx) omega)->((in ((setadjoin Xx) Xx)) omega))):Prop.
% Definition peano0notS:=(forall (Xx:fofType), (((in Xx) omega)->(not (((eq fofType) (omegaS Xx)) emptyset)))):Prop.
% Parameter peano3:(fofType->(fofType->(fofType->Prop))).
% Parameter peano4:(fofType->(fofType->(fofType->Prop))).
% Parameter peano5:(fofType->(fofType->(fofType->Prop))).
% Definition peanoSinj:=(forall (Xx:fofType), (((in Xx) omega)->(forall (Xy:fofType), (((in Xy) omega)->((((eq fofType) (omegaS Xx)) (omegaS Xy))->(((eq fofType) Xx) Xy)))))):Prop.
% Parameter peano:(fofType->(fofType->(fofType->Prop))).
% Definition transitiveset:=(fun (A:fofType)=> (forall (X:fofType), (((in X) A)->((subset X) A)))):(fofType->Prop).
% Definition transitivesetOp1:=(forall (X:fofType), ((transitiveset X)->(forall (A:fofType), (((in A) X)->((subset A) X))))):Prop.
% Trying to prove (setextAx->(emptysetAx->(setadjoinAx->(powersetAx->(setunionAx->(omega0Ax->(omegaSAx->(omegaIndAx->(replAx->(foundationAx->(wellorderingAx->(descrp->(dsetconstrI->(dsetconstrEL->(dsetconstrER->(exuE1->(prop2setE->(emptysetE->(emptysetimpfalse->(notinemptyset->(exuE3e->(setext->(emptyI->(noeltsimpempty->(setbeta->(nonemptyE1->(nonemptyI->(nonemptyI1->(setadjoinIL->(emptyinunitempty->(setadjoinIR->(setadjoinE->(setadjoinOr->(setoftrueEq->(powersetI->(emptyinPowerset->(emptyInPowerset->(powersetE->(setunionI->(setunionE->(subPowSU->(exuE2->(nonemptyImpWitness->(uniqinunit->(notinsingleton->(eqinunit->(singletonsswitch->(upairsetE->(upairsetIL->(upairsetIR->(emptyE1->(vacuousDall->(quantDeMorgan1->(quantDeMorgan2->(quantDeMorgan3->(quantDeMorgan4->(prop2setI->(prop2set2propI->(notdexE->(notdallE->(exuI1->(exuI3->(exuI2->(inCongP->(in__Cong->(exuE3u->(exu__Cong->(emptyset__Cong->(setadjoin__Cong->(powerset__Cong->(setunion__Cong->(omega__Cong->(exuEu->(descr__Cong->(dsetconstr__Cong->(subsetI1->(eqimpsubset2->(eqimpsubset1->(subsetI2->(emptysetsubset->(subsetE->(subsetE2->(notsubsetI->(notequalI1->(notequalI2->(subsetRefl->(subsetTrans->(setadjoinSub->(setadjoinSub2->(subset2powerset->(setextsub->(subsetemptysetimpeq->(powersetI1->(powersetE1->(inPowerset->(powersetsubset->(sepInPowerset->(sepSubset->(binunionIL->(upairset2IR->(binunionIR->(binunionEcases->(binunionE->(binunionLsub->(binunionRsub->(binintersectI->(binintersectSubset5->(binintersectEL->(binintersectLsub->(binintersectSubset2->(binintersectSubset3->(binintersectER->(disjointsetsI1->(binintersectRsub->(binintersectSubset4->(binintersectSubset1->(bs114d->(setminusI->(setminusEL->(setminusER->(setminusSubset2->(setminusERneg->(setminusELneg->(setminusILneg->(setminusIRneg->(setminusLsub->(setminusSubset1->(symdiffE->(symdiffI1->(symdiffI2->(symdiffIneg1->(symdiffIneg2->(secondinupair->(setukpairIL->(setukpairIR->(kpairiskpair->(kpairp->(singletonsubset->(singletoninpowerset->(singletoninpowunion->(upairset2E->(upairsubunion->(upairinpowunion->(ubforcartprodlem1->(ubforcartprodlem2->(ubforcartprodlem3->(cartprodpairin->(cartprodmempair1->(cartprodmempair->(setunionE2->(setunionsingleton1->(setunionsingleton2->(setunionsingleton->(singletonprop->(ex1E1->(ex1I->(ex1I2->(singletonsuniq->(setukpairinjL1->(kfstsingleton->(theprop->(kfstpairEq->(cartprodfstin->(setukpairinjL2->(setukpairinjL->(setukpairinjR11->(setukpairinjR12->(setukpairinjR1->(upairequniteq->(setukpairinjR2->(setukpairinjR->(ksndsingleton->(ksndpairEq->(kpairsurjEq->(cartprodsndin->(cartprodpairmemEL->(cartprodpairmemER->(cartprodmempaircEq->(cartprodfstpairEq->(cartprodsndpairEq->(cartprodpairsurjEq->(dpsetconstrI->(dpsetconstrSub->(setOfPairsIsBReln->(dpsetconstrERa->(dpsetconstrEL1->(dpsetconstrEL2->(dpsetconstrER->(funcImageSingleton->(apProp->(app->(infuncsetfunc->(ap2p->(funcinfuncset->(lamProp->(lamp->(lam2p->(brelnall1->(brelnall2->(ex1E2->(funcGraphProp1->(funcGraphProp3->(funcGraphProp2->(funcextLem->(funcGraphProp4->(subbreln->(eqbreln->(funcext->(funcext2->(ap2apEq1->(ap2apEq2->(beta1->(eta1->(lam2lamEq->(beta2->(eta2->(iffalseProp1->(iffalseProp2->(iftrueProp1->(iftrueProp2->(ifSingleton->(ifp->(theeq->(iftrue->(iffalse->(iftrueorfalse->(binintersectT_lem->(binunionT_lem->(powersetT_lem->(setminusT_lem->(complementT_lem->(setextT->(subsetTI->(powersetTI1->(powersetTE1->(complementTI1->(complementTE1->(binintersectTELcontra->(binintersectTERcontra->(contrasubsetT->(contrasubsetT1->(contrasubsetT2->(contrasubsetT3->(doubleComplementI1->(doubleComplementE1->(doubleComplementSub1->(doubleComplementSub2->(double
% EOF
%------------------------------------------------------------------------------